The algebraic structure of semi-brace
Paola Stefanelli
Università del Salento
Spa, 22 June 2017
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Notations
We introduce the semi-brace, an algebraic structure that allows us to obtainleft non-degenerate solutions of the Yang-Baxter equation.
If X is a set, a (set-theoretical) solution of the Yang-Baxter equationr : X × X → X × X is a map such that the well-known braid equation
r1r2r1 = r2r1r2
is satis�ed, where r1 = r × idX and r2 = idX ×r .
If a, b ∈ X , we denote r (a, b) = (λa (b) , ρb (a)) where λa, ρb are maps from Xinto itself.
In particular, we say that r is left (right, resp.) non-degenerate if λa (ρa, resp.)is bijective, for every a ∈ X . Moreover, r is non-degenerate if r is both left andright non-degenerate.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 1 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Skew Braces
In 2016, Guarnieri and Vendramin introduced a new algebraic structure, theskew braces, in order to obtain bijective solutions not necessarily involutive.
De�nition
Let B be a set with two operations + and ◦ such that (B,+) and (B, ◦) aregroups. We say that (B,+, ◦) is a skew (left) brace if
a ◦ (b + c) = a ◦ b − a+ a ◦ c,
holds for all a, b, c ∈ B, where −a is the inverse of a with respect to +.
We may prove that the identity 0 of (B,+) is also the identity of (B, ◦).
If the group (B,+) is abelian, then (B,+, ◦) is a brace, the algebraic structureintroduced by Rump, in the reformulation provided by Cedó, Jespers andOkni«ski.
Clearly, every brace is a skew brace. Further, if (B,+) is a group and we seta ◦ b := a+ b, for all a, b ∈ B, then (B,+, ◦) is a skew brace, that we call zeroskew brace. If (B,+) is a non-abelian group then B is a skew brace that is nota brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 2 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Semi-bracesWe introduced a generalization of skew braces.
De�nition (F. Catino, I. Colazzo, and P.S., J. Algebra, 2017)
Let B be a set with two operations + and ◦ such that (B,+) is a leftcancellative semigroup and (B, ◦) is a group. We say that (B,+, ◦) is a(left) semi-brace if
a ◦ (b + c) = a ◦ b + a ◦(a− + c
),
holds for all a, b, c ∈ B, where a− is the inverse of a with respect to ◦.
Note that, if B is a skew brace, then it is a semi-brace. In fact
I (B,+) is a group and, in particular, a left cancellative semigroup;
I if a, b, c ∈ B then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ a− − a+ a ◦ c= a ◦ b + 0− a+ a ◦ c = a ◦ b − a+ a ◦ c= a ◦ (b + c)
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 3 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I if a, b, c ∈ B, then a+ (b + c) = a+ (c ◦ f (b)) = c ◦ f (b) ◦ f (a) =
c ◦ f (b) ◦ f 2 (a) = c ◦ f (b ◦ f (a)) = (a+ b) + cI (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I if a, b, c ∈ B, such that a+ b = a+ c, then b ◦ f (a) = c ◦ f (a), i.e., b = c
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples of semi-braces
1. If (E , ◦) is a group, then (E ,+, ◦), where a+ b = b, for all a, b ∈ E is asemi-brace. In fact,
I (E ,+) is a left cancellative semigroup;
I if a, b, c ∈ B, then a ◦ b + a ◦(a− + c
)= a ◦ c = a ◦ (b + c).
We call this semi-brace the trivial semi-brace.
2. If (B, ◦) is a group and f is an endomorphism of (B, ◦) such that f 2 = f .Set
a+ b := b ◦ f (a) ,
for all a, b ∈ B, then (B,+, ◦) is a semi-brace. In fact,I + is associative;
I (B,+) is left cancellative;
I if a, b, c ∈ B, then
a ◦ b + a ◦(a− + c
)= a ◦ b + a ◦ c ◦ f
(a−
)= a ◦ c ◦ f
(a−
)◦ f (a ◦ b)
= a ◦ c ◦ f(a− ◦ a ◦ b
)= a ◦ c ◦ f (b)
= a ◦ (b + c) .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 4 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - I
Note that, if B is a semi-brace and 0 is the identity of (B, ◦), then 0 is a leftidentity (and, also, an idempotent) of (B,+). In fact if a ∈ B, then
0+ a = 0 ◦ (0+ a) = 0 ◦ 0+ 0 ◦ (0+ a) = 0+ 0+ a
and, by left cancellativity, we have that a = 0+ a.
Recall that a left cancellative semigroup B is a right group if and only if for allx , y ∈ B there exists t ∈ B such that x + t = y .
If B is a semi-brace, x , y ∈ B and we set t := x ◦(x− + x− ◦ y
), then
x + t = x + x ◦(x− + x− ◦ y
)= x ◦
(0+ x− ◦ y
)= x ◦ x− ◦ y = y .
Hence, the additive structure (B,+) is a right group.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 5 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The additive structure - II
Moreover, it is well-known that if B is a right group, E is the set ofidempotents, then Ge := B + e, for every e ∈ E , is a group and B = Ge + E
In particular, if B is a semi-brace and E is the set of idempotents of (B,+),then the identity 0 of the group (B, ◦) lies in E . Therefore G := B + 0 is agroup with respect to the sum and
B = G + E .
In addition, we may prove that (G , ◦) and (E , ◦) are groups and so (G ,+, ◦) isa skew brace and (E ,+, ◦) is a trivial semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 6 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Examples
I If (E ,+, ◦) is a trivial semi-brace, then the set of idempotents of (E ,+) isE and the group G = {0}.
I If (B,+, ◦) is the semi-brace where f : B → B is an endomorphism of thegroup (B, ◦), f 2 = f and a+ b = b ◦ f (a), for all a, b ∈ B. The set ofidempotents of (B,+) is ker f and the group G := B + 0 is Im f . In fact,
x ∈ E ⇐⇒ x + x = x ⇐⇒ x ◦ f (x) = x ⇐⇒ f (x) = 0
⇐⇒ x ∈ ker f .
and
x ∈ G ⇐⇒ x + 0 = x ⇐⇒ 0 ◦ f (x) = x ⇐⇒ x ∈ Im f .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 7 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The multiplicative group
Let B be a semi-brace, E the set of idempotents of (B,+), and G := B + 0.
I Clearly, we have that G ∩ E = {0}.
I Further, B = G ◦ E . In fact, if b ∈ B, then there exist g ∈ G and e ∈ Esuch that b = g + e and so
b = g︸︷︷︸∈G
◦ g− ◦ (g + e)︸ ︷︷ ︸∈E
.
In fact,
g− ◦ (g + e) + g− ◦ (g + e) = g− ◦ (g + e + e) = g− ◦ (g + e) ,
i.e., g− ◦ (g + e) ∈ E .
I Since (G , ◦) and (E , ◦) are groups, we have that (B, ◦) is the matchedproduct of the groups (G , ◦) and (E , ◦).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 8 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a skew brace
Guarnieri and Vendramin give the following de�nition of ideal for a skew brace.
De�nition
Let B be a skew brace. A subset I of B is said an ideal if
I I is a normal subgroup of (B, ◦);I I is a normal subgroup of (B,+);
I λa (I ) ⊆ I , for every a ∈ B, where λa (b) := −a+ a ◦b, for all a, b ∈ B.
In particular, if B is a brace, the second condition follows by the �rst and thirdones.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 9 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - I
De�nition (F. Catino, I. Colazzo, P. S., J. Algebra, 2017)
Let B be a semi-brace, E the set of idempotents of (B,+), G := B+0. Wesay that a subsemigroup I of (B,+) is an ideal if
I I is a normal subgroup of (B, ◦);I I ∩ G is a normal subgroup of (G ,+);
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ;
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G .
Where, if a ∈ B, λa : B → B and ρa : B → B are de�ned respectively by
λa (b) := a ◦(a− + b
)and ρa (b) :=
(b− + a
)− ◦ afor every b ∈ B.
As we expect, B and {0} are ideals of B that we call the trivial ideals of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 10 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; λg (0) = −g + g ◦ 0 = 0 ∈ I
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - I
If B is a skew brace and I is an ideal of B, then I is an ideal of B reviewed as asemi-brace. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G is a normal subgroup of (G ,+); 3
I λg (e) ∈ I , for all g ∈ G and e ∈ I ∩ E ; 3
I ρb (n) ∈ I , for all b ∈ B and n ∈ I ∩ G . 3
ρb (n) =(n− + b
)− ◦ b =(b− ◦
(n− + b
))−=(b− ◦
(b − b + n− + b
))−=(b− ◦ b − b− + b− ◦
(−b + n− + b
))−= (λb−(−b + n− + b︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 11 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Comparison between the two de�nitions of ideal - II
Conversely, if B a skew brace and I is an ideal of B reviewed as a semi-brace,then I is an ideal of the skew brace B. In fact,
I I is a normal subgroup of (B, ◦); 3
I I is a normal subgroup of (B,+); 3
I λa (I ) ⊆ I , for every a ∈ B. 3
λa (x) = a ◦(a− + x
)=((
a− + x)− ◦ a−)− =
((a− + x − a− + a−
)◦ a−
)−= (ρa−(a− + x − a−︸ ︷︷ ︸
∈I
))− ∈ I .
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 12 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Remark
In the case of a semi-brace that is not a skew brace, there is a di�erence withrespect to the ideal of a skew brace.
Let (B, ◦) be a group that is not simple, reviewed as trivial semi-brace. If I is anon-trivial normal subgroup of (B, ◦), then I is an ideal. In fact,
I I is a normal subgroup of (B, ◦); 3
I I ∩ G = {0} is a normal subgroup of (G ,+); 3
I λ0 (e) = 0 ◦ (0+ e) = e ∈ I , for every e ∈ I ∩ E = I ; 3
I ρb (0) = (0+ b)− ◦ b = b− ◦ b = 0 ∈ I , for every b ∈ B. 3
But if a ∈ B \ I , then
λa (0) = a ◦(a− + 0
)= a ◦ 0 = a /∈ I ,
i.e., I is not λa - invariant, nevertheless I is an ideal of B.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 13 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
Ideals of a semi-brace - II
Proposition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace and I is an ideal of B, then the relation ∼I on B given
by
∀ x , y ∈ B, x ∼I y ⇐⇒ y− ◦ x ∈ I
is a congruence of B.
Further, if B is a semi-brace and I is an ideal, then the quotient structure B/Iof B with respect to the relation ∼I is a right group with respect to the sum.
Therefore the quotient structure B/I is a semi-brace.
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 14 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
De�nitions and examples Additive and multiplicative structures Ideals and quotient structures
The socle
Guarnieri and Vendramin introduced the socle for skew braces, asgeneralization of that classical for braces.
De�nition
Let B be a skew brace. Then the ideal de�ned by
S (B) := {a | a ∈ B, ∀ b ∈ B a ◦ b = a+ b, b + b ◦ a = b ◦ a+ b}
is said the socle of B.
We may generalize this de�nition for semi-braces in the following way:
De�nition (F. Catino, I. Colazzo, P.S., J. Algebra, 2017)
If B is a semi-brace, 0 is the identity of (B, ◦) and G := B + 0, then we callthe set given by
Soc (B) = {a | a ∈ G , ∀ b ∈ B a ◦ b = a+ b, −a+ b + a = b + 0} .
the socle of the semi-brace B.
If B is a skew brace, then S (B) = Soc (B).
P. Stefanelli (UniSalento) The algebraic structure of semi-brace 15 / 15
Thanks for your attention!