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Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) [email protected] In this talk, we introduce and investigate the notions of semibialgebras and Hopf semialgebras over semirings. We also investigate several related categories of Doi- Koppinen semimodules.
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Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) [email protected] In this talk, we introduce and investigate

Jul 13, 2020

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Page 1: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

Hopf semialgebrasJawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia)

[email protected]

In this talk, we introduce and investigate the notions of semibialgebras and Hopfsemialgebras over semirings. We also investigate several related categories of Doi-Koppinen semimodules.

Page 2: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Hopf Semialgebras

Hopf algebras and tensor categoriesUniversity of Almerıa (Spain)

July 4 – 8, 2011

Jawad Abuhlail

King Fahd University of Petroleum & Minerals

July 5, 2011Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 3: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Outline

1 Preliminaries

2 The category SS

3 Hopf Semialgebras

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 4: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 5: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 6: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }

R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 7: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)

R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 8: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ring

S : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 9: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiring

SS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 10: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodules

SCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 11: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 12: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Preliminaries

AbstractIn this talk, we introduce and investigate the notions ofsemibialgebras and Hopf semialgebras over semirings. We alsoprove the Fundamental Theorem of Hopf Semialgebras.

NotationN0 := {0, 1, 2, 3, · · · }R+ := [0,∞)R : ringS : semiringSS: the category of left S-semimodulesSCS: the category of cancellative left S-semimodules

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 13: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 14: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 15: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 16: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring;

R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 17: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;

(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 18: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);

A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 19: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.

(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 20: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semirings

Semirings are, roughly speaking, rings withoutsubtraction, i.e. (S ,+, 0) is a commutative monoid (notnecessarily a group) with s · 0 = 0 = 0 · s ∀ s ∈ S .

Semirings have numerous applications, e.g. Golan (1999,2003), Hebisch & Weinert (1998), Kuich (1986).

A semiring is the best structure which includes both ringsand bounded distributive lattices (Vandiver 1934).

Example

(N0,+, .) is a semiring; R+ and Q+ are semifields;(ideal(R),+, .) is a semiring (Dedekind 1894);A domain D is Prufer ⇔ (ideal(D),+,∩) is a semiring.(L,∨,∧) is a b.d. lattice having unique minimal & uniquemaximal ⇔ L is a comm. idempotent simple semiring.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 21: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Examples & Remarks

The semiring Rmax := (R ∪ {−∞},max ,+), which is aidempotent dequantization of R+, plays an important rolein Tropical Geometry and Idempotent Analysis (e.g.Mikhalkin 2006; Litvinov & Maslov 2005, Litvinov 2010).

The semiring (N ∪ {∞},min,+) plays an important rulein Automata Theory (Conway 1971; Eilenberg 1974;Salomaa & Soittola 1978).

The category F of finite sets with partial bijectionsendowed with ⊕ (disjoint union) and ⊗ (cartesianproduct) is a semiring (Pena & Lorscheid 2009).

Semirings ↔ additive algebraic monads on Set (Durov2007).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 22: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Examples & Remarks

The semiring Rmax := (R ∪ {−∞},max ,+), which is aidempotent dequantization of R+, plays an important rolein Tropical Geometry and Idempotent Analysis (e.g.Mikhalkin 2006; Litvinov & Maslov 2005, Litvinov 2010).

The semiring (N ∪ {∞},min,+) plays an important rulein Automata Theory (Conway 1971; Eilenberg 1974;Salomaa & Soittola 1978).

The category F of finite sets with partial bijectionsendowed with ⊕ (disjoint union) and ⊗ (cartesianproduct) is a semiring (Pena & Lorscheid 2009).

Semirings ↔ additive algebraic monads on Set (Durov2007).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 23: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Examples & Remarks

The semiring Rmax := (R ∪ {−∞},max ,+), which is aidempotent dequantization of R+, plays an important rolein Tropical Geometry and Idempotent Analysis (e.g.Mikhalkin 2006; Litvinov & Maslov 2005, Litvinov 2010).

The semiring (N ∪ {∞},min,+) plays an important rulein Automata Theory (Conway 1971; Eilenberg 1974;Salomaa & Soittola 1978).

The category F of finite sets with partial bijectionsendowed with ⊕ (disjoint union) and ⊗ (cartesianproduct) is a semiring (Pena & Lorscheid 2009).

Semirings ↔ additive algebraic monads on Set (Durov2007).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 24: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Examples & Remarks

The semiring Rmax := (R ∪ {−∞},max ,+), which is aidempotent dequantization of R+, plays an important rolein Tropical Geometry and Idempotent Analysis (e.g.Mikhalkin 2006; Litvinov & Maslov 2005, Litvinov 2010).

The semiring (N ∪ {∞},min,+) plays an important rulein Automata Theory (Conway 1971; Eilenberg 1974;Salomaa & Soittola 1978).

The category F of finite sets with partial bijectionsendowed with ⊕ (disjoint union) and ⊗ (cartesianproduct) is a semiring (Pena & Lorscheid 2009).

Semirings ↔ additive algebraic monads on Set (Durov2007).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 25: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semimodules

A left semimodule over a semiring is, roughly speaking, aleft module M without subtraction ( (M ,+, 0) is acommutative monoid with0S m = 0M = s 0M ∀ s ∈ S ,m ∈ M .

For the foundations of the theory of semimodules oversemirings we refer to the fundamental series of papers byM. Takahashi (1979 - 1985) in addition to Golan’s books(1999, 2002).

A comprehensive literature review on semirings and theirapplications is provided by K. Głazek (2002).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 26: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semimodules

A left semimodule over a semiring is, roughly speaking, aleft module M without subtraction ( (M ,+, 0) is acommutative monoid with0S m = 0M = s 0M ∀ s ∈ S ,m ∈ M .

For the foundations of the theory of semimodules oversemirings we refer to the fundamental series of papers byM. Takahashi (1979 - 1985) in addition to Golan’s books(1999, 2002).

A comprehensive literature review on semirings and theirapplications is provided by K. Głazek (2002).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 27: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semimodules

A left semimodule over a semiring is, roughly speaking, aleft module M without subtraction ( (M ,+, 0) is acommutative monoid with0S m = 0M = s 0M ∀ s ∈ S ,m ∈ M .

For the foundations of the theory of semimodules oversemirings we refer to the fundamental series of papers byM. Takahashi (1979 - 1985) in addition to Golan’s books(1999, 2002).

A comprehensive literature review on semirings and theirapplications is provided by K. Głazek (2002).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 28: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Factorization StructuresDefinition(Adamek, Herrlich and Strecker 2004) Let E ⊆ Epi andM ⊆Mono. We call (E,M) a factorization structure formorphisms in C provided that

1 E and M are closed under composition withisomorphisms.

2 C has (E,M)-factorizations, i.e. each morphism h in Chas a factorization h = m ◦ e with m ∈M and e ∈ E.

3 C has the unique (E,M)-diagonalization property

A e //

f��

B

g��∃ ! d

xxC m

// D

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 29: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Factorization StructuresDefinition(Adamek, Herrlich and Strecker 2004) Let E ⊆ Epi andM ⊆Mono. We call (E,M) a factorization structure formorphisms in C provided that

1 E and M are closed under composition withisomorphisms.

2 C has (E,M)-factorizations, i.e. each morphism h in Chas a factorization h = m ◦ e with m ∈M and e ∈ E.

3 C has the unique (E,M)-diagonalization property

A e //

f��

B

g��∃ ! d

xxC m

// D

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 30: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Factorization StructuresDefinition(Adamek, Herrlich and Strecker 2004) Let E ⊆ Epi andM ⊆Mono. We call (E,M) a factorization structure formorphisms in C provided that

1 E and M are closed under composition withisomorphisms.

2 C has (E,M)-factorizations, i.e. each morphism h in Chas a factorization h = m ◦ e with m ∈M and e ∈ E.

3 C has the unique (E,M)-diagonalization property

A e //

f��

B

g��∃ ! d

xxC m

// D

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 31: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)

is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 32: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)

monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 33: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphisms

regular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 34: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphisms

is complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 35: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)

is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 36: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)

has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 37: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 38: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... good propertiesTakahashi 1982

is a variety (i.e. an HSP class in the sense of UA)is Barr-exact (1971) (with a(RegEpi,Mono)-factorization structure)monomorphisms = injective morphismsregular epimorphisms = surjective morphismsis complete (i.e. has equalizers & products)is cocomplete (i.e. has coequalizers & coproducts)has kernels and cokernels.

Example

An epimorphism which is not surjective:

h : N0 × N0 → N0 × N0, (n,m) 7→ (2n + m, n).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 39: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... bad properties

Observationsnot monoidal (with Takahashi’s tensor product:S �S M ' c(M))

neither left nor right closed;

not subtractive (in the sense of A. Ursini 1994)

not exact (in the sense of Puppe: 1962; a pointedcategory with a(NormalEpi,NormalMono)-factorization structure)

not homological (in the sense of Borceux & Bourn: 2004;a category which is pointed, regular & protomodular)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 40: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... bad properties

Observationsnot monoidal (with Takahashi’s tensor product:S �S M ' c(M))

neither left nor right closed;

not subtractive (in the sense of A. Ursini 1994)

not exact (in the sense of Puppe: 1962; a pointedcategory with a(NormalEpi,NormalMono)-factorization structure)

not homological (in the sense of Borceux & Bourn: 2004;a category which is pointed, regular & protomodular)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 41: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... bad properties

Observationsnot monoidal (with Takahashi’s tensor product:S �S M ' c(M))

neither left nor right closed;

not subtractive (in the sense of A. Ursini 1994)

not exact (in the sense of Puppe: 1962; a pointedcategory with a(NormalEpi,NormalMono)-factorization structure)

not homological (in the sense of Borceux & Bourn: 2004;a category which is pointed, regular & protomodular)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 42: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... bad properties

Observationsnot monoidal (with Takahashi’s tensor product:S �S M ' c(M))

neither left nor right closed;

not subtractive (in the sense of A. Ursini 1994)

not exact (in the sense of Puppe: 1962; a pointedcategory with a(NormalEpi,NormalMono)-factorization structure)

not homological (in the sense of Borceux & Bourn: 2004;a category which is pointed, regular & protomodular)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 43: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

The category SS ... bad properties

Observationsnot monoidal (with Takahashi’s tensor product:S �S M ' c(M))

neither left nor right closed;

not subtractive (in the sense of A. Ursini 1994)

not exact (in the sense of Puppe: 1962; a pointedcategory with a(NormalEpi,NormalMono)-factorization structure)

not homological (in the sense of Borceux & Bourn: 2004;a category which is pointed, regular & protomodular)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 44: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Subtractive Subsemimodules

DefinitionLet M be a semimodule.

A non-empty subset L ⊂ M is said to be subtractive iff:l + m, l ∈ L⇒ m ∈ L for all m, l ∈ M .

The subtractive closure of a subsemimodule L ¬S M is

L := {m ∈ M |m + l1 = l2 for some l1, l2 ∈ L}.

LemmaThe following are equivalent for an S-semimodule M:

1 M is cancellative (i.e. m1 + m = m2 + m⇒ m1 = m2);2 W := {(m,m) | m ∈ M} ⊂ M ×M is subtractive;3 ξ : M → M∆ is injective, where M∆ := (M ×M)/W .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 45: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Subtractive Subsemimodules

DefinitionLet M be a semimodule.

A non-empty subset L ⊂ M is said to be subtractive iff:l + m, l ∈ L⇒ m ∈ L for all m, l ∈ M .The subtractive closure of a subsemimodule L ¬S M is

L := {m ∈ M |m + l1 = l2 for some l1, l2 ∈ L}.

LemmaThe following are equivalent for an S-semimodule M:

1 M is cancellative (i.e. m1 + m = m2 + m⇒ m1 = m2);2 W := {(m,m) | m ∈ M} ⊂ M ×M is subtractive;3 ξ : M → M∆ is injective, where M∆ := (M ×M)/W .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 46: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Subtractive Subsemimodules

DefinitionLet M be a semimodule.

A non-empty subset L ⊂ M is said to be subtractive iff:l + m, l ∈ L⇒ m ∈ L for all m, l ∈ M .The subtractive closure of a subsemimodule L ¬S M is

L := {m ∈ M |m + l1 = l2 for some l1, l2 ∈ L}.

LemmaThe following are equivalent for an S-semimodule M:

1 M is cancellative (i.e. m1 + m = m2 + m⇒ m1 = m2);

2 W := {(m,m) | m ∈ M} ⊂ M ×M is subtractive;3 ξ : M → M∆ is injective, where M∆ := (M ×M)/W .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 47: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Subtractive Subsemimodules

DefinitionLet M be a semimodule.

A non-empty subset L ⊂ M is said to be subtractive iff:l + m, l ∈ L⇒ m ∈ L for all m, l ∈ M .The subtractive closure of a subsemimodule L ¬S M is

L := {m ∈ M |m + l1 = l2 for some l1, l2 ∈ L}.

LemmaThe following are equivalent for an S-semimodule M:

1 M is cancellative (i.e. m1 + m = m2 + m⇒ m1 = m2);2 W := {(m,m) | m ∈ M} ⊂ M ×M is subtractive;

3 ξ : M → M∆ is injective, where M∆ := (M ×M)/W .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 48: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Subtractive Subsemimodules

DefinitionLet M be a semimodule.

A non-empty subset L ⊂ M is said to be subtractive iff:l + m, l ∈ L⇒ m ∈ L for all m, l ∈ M .The subtractive closure of a subsemimodule L ¬S M is

L := {m ∈ M |m + l1 = l2 for some l1, l2 ∈ L}.

LemmaThe following are equivalent for an S-semimodule M:

1 M is cancellative (i.e. m1 + m = m2 + m⇒ m1 = m2);2 W := {(m,m) | m ∈ M} ⊂ M ×M is subtractive;3 ξ : M → M∆ is injective, where M∆ := (M ×M)/W .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 49: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Cancellative Semimodules

DefinitionLet M be an S-semimodule and consider the S-congruencerelation

m[≡]{0}m′ ⇔ m + m′′ = m′ + m′′; m′′ ∈ M .

Then we have a cancellative S-semimodule

c(M) := M/[≡]{0} = {[m]{0} : m ∈ M}. (1)

Moreover, we have a canonical surjection cM : M → c(M) with

δ(M) := Ker(cM) = {m ∈ M | m + m′ = m′; m′ ∈ M}.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 50: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Takahashi’s Tensor Product 1982

Universal Property

Let M be a right S-semimodule and N a left S-semimodule.For every cancellative commutative monoid G :

M × N

τ ′

τ

||

M �S N ∃ τ // G

M �S SϑrM' c(M) and S �S N

ϑlN' c(N)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 51: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Takahashi’s Tensor Product 1982

Universal Property

Let M be a right S-semimodule and N a left S-semimodule.For every cancellative commutative monoid G :

M × N

τ ′

τ

||

M �S N ∃ τ // G

M �S SϑrM' c(M) and S �S N

ϑlN' c(N)

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 52: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Katsov ... 1997

Universal Property

Let M be a right S-semimodule and N a left S-semimodule.The universal property holds for any commutative monoid G :

M × N

τ ′

τ

||

M ⊗S N ∃ τ // G

M ⊗S SϑrM' M and S ⊗S N

ϑlN' N

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 53: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Katsov ... 1997

Universal Property

Let M be a right S-semimodule and N a left S-semimodule.The universal property holds for any commutative monoid G :

M × N

τ ′

τ

||

M ⊗S N ∃ τ // G

M ⊗S SϑrM' M and S ⊗S N

ϑlN' N

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 54: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exactness in Non-exact Categories ... (underpreparation)

DefinitionLet C be any pointed category. We call a sequenceA f−→ B

g−→ C exact ifff = ker(g) ◦ f ′, where (f ′, ker(g)) ∈ E×M;g = g ′′ ◦ coker(f ), where (coker(f ), g ′′) ∈ E×M.

Example

Let C be an exact category. The following are equivalent:1 A f−→ B

g−→ C is exact.2 f = ker(g) ◦ f ′, where f ′ ∈ E := NormalEpi;3 g = g ′′ ◦ coker(f ), where g ′′ ∈M := NormalMono.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 55: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exactness in Non-exact Categories ... (underpreparation)

DefinitionLet C be any pointed category. We call a sequenceA f−→ B

g−→ C exact ifff = ker(g) ◦ f ′, where (f ′, ker(g)) ∈ E×M;g = g ′′ ◦ coker(f ), where (coker(f ), g ′′) ∈ E×M.

Example

Let C be an exact category. The following are equivalent:1 A f−→ B

g−→ C is exact.

2 f = ker(g) ◦ f ′, where f ′ ∈ E := NormalEpi;3 g = g ′′ ◦ coker(f ), where g ′′ ∈M := NormalMono.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 56: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exactness in Non-exact Categories ... (underpreparation)

DefinitionLet C be any pointed category. We call a sequenceA f−→ B

g−→ C exact ifff = ker(g) ◦ f ′, where (f ′, ker(g)) ∈ E×M;g = g ′′ ◦ coker(f ), where (coker(f ), g ′′) ∈ E×M.

Example

Let C be an exact category. The following are equivalent:1 A f−→ B

g−→ C is exact.2 f = ker(g) ◦ f ′, where f ′ ∈ E := NormalEpi;

3 g = g ′′ ◦ coker(f ), where g ′′ ∈M := NormalMono.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 57: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exactness in Non-exact Categories ... (underpreparation)

DefinitionLet C be any pointed category. We call a sequenceA f−→ B

g−→ C exact ifff = ker(g) ◦ f ′, where (f ′, ker(g)) ∈ E×M;g = g ′′ ◦ coker(f ), where (coker(f ), g ′′) ∈ E×M.

Example

Let C be an exact category. The following are equivalent:1 A f−→ B

g−→ C is exact.2 f = ker(g) ◦ f ′, where f ′ ∈ E := NormalEpi;3 g = g ′′ ◦ coker(f ), where g ′′ ∈M := NormalMono.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 58: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exact SequencesDefinitionWe call an S-linear morphism f : M → N:

subtractive (i -regular: Takahashi) iff f (M) = f (M);steady (k-regular: Takahashi) iff

f (m) = f (m′)⇒ m+k = m′+k ′ for some k , k ′ ∈ Ker(f ).

Exact Sequences

We call a sequence of S-semimodules L f→ Mg→ N :

exact iff f (L) = Ker(g) and g is steady;proper-exact iff f (L) = Ker(g) (exact in Patchkoria 2003);semi-exact iff f (L) = Ker(g) (exact in Takahashi 1981);weakly-exact iff f (L) = Ker(g) and g is steady (PD2006).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 59: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exact SequencesDefinitionWe call an S-linear morphism f : M → N:

subtractive (i -regular: Takahashi) iff f (M) = f (M);

steady (k-regular: Takahashi) iff

f (m) = f (m′)⇒ m+k = m′+k ′ for some k , k ′ ∈ Ker(f ).

Exact Sequences

We call a sequence of S-semimodules L f→ Mg→ N :

exact iff f (L) = Ker(g) and g is steady;proper-exact iff f (L) = Ker(g) (exact in Patchkoria 2003);semi-exact iff f (L) = Ker(g) (exact in Takahashi 1981);weakly-exact iff f (L) = Ker(g) and g is steady (PD2006).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 60: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exact SequencesDefinitionWe call an S-linear morphism f : M → N:

subtractive (i -regular: Takahashi) iff f (M) = f (M);steady (k-regular: Takahashi) iff

f (m) = f (m′)⇒ m+k = m′+k ′ for some k , k ′ ∈ Ker(f ).

Exact Sequences

We call a sequence of S-semimodules L f→ Mg→ N :

exact iff f (L) = Ker(g) and g is steady;proper-exact iff f (L) = Ker(g) (exact in Patchkoria 2003);semi-exact iff f (L) = Ker(g) (exact in Takahashi 1981);weakly-exact iff f (L) = Ker(g) and g is steady (PD2006).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 61: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exact SequencesDefinitionWe call an S-linear morphism f : M → N:

subtractive (i -regular: Takahashi) iff f (M) = f (M);steady (k-regular: Takahashi) iff

f (m) = f (m′)⇒ m+k = m′+k ′ for some k , k ′ ∈ Ker(f ).

Exact Sequences

We call a sequence of S-semimodules L f→ Mg→ N :

exact iff f (L) = Ker(g) and g is steady;

proper-exact iff f (L) = Ker(g) (exact in Patchkoria 2003);semi-exact iff f (L) = Ker(g) (exact in Takahashi 1981);weakly-exact iff f (L) = Ker(g) and g is steady (PD2006).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 62: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Exact SequencesDefinitionWe call an S-linear morphism f : M → N:

subtractive (i -regular: Takahashi) iff f (M) = f (M);steady (k-regular: Takahashi) iff

f (m) = f (m′)⇒ m+k = m′+k ′ for some k , k ′ ∈ Ker(f ).

Exact Sequences

We call a sequence of S-semimodules L f→ Mg→ N :

exact iff f (L) = Ker(g) and g is steady;proper-exact iff f (L) = Ker(g) (exact in Patchkoria 2003);semi-exact iff f (L) = Ker(g) (exact in Takahashi 1981);weakly-exact iff f (L) = Ker(g) and g is steady (PD2006).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 63: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

SemialgebrasAssumption

From now on, S is a commutative semiring, so that (SS,⊗, S)is a symmetric monoidal category.

DefinitionWith an S-semialgebra, we mean a triple (A, µA, ηA), where Ais a S-semimodule and

µ : A⊗S A→ A and η : S → A

are S-linear morphisms such that

µA ◦ (µA ⊗S idA) = µA ◦ (idA ⊗S µA);

µA ◦ (ηA ⊗S idA) = ϑlA & µA ◦ (idA ⊗S ηA) = ϑrA.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 64: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semicoalgebras ... (Brussels, 2008)DefinitionAn S-semicoalgebra is an S-semimodule C associated with

∆C : C → C �S C and εC : C → S ,

C∆C //

∆C

��

C �S C

idC�S∆C

��C �S C

∆C�S idC// C �S C �S C

C �S C

εC�S idC��

C∆Coo ∆C //

cC��

C �S C

idC�SεC��

S �S CϑlC

// c(C ) C �S SϑrC

oo

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 65: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semicoalgebras ... (Almeria, 2011)DefinitionAn S-semicoalgebra is an S-semimodule C associated with

∆C : C → C ⊗S C and εC : C → S ,

C∆C //

∆C

��

C ⊗S C

idC⊗S∆C

��C ⊗S C

∆C⊗S idC// C ⊗S C ⊗S C

C ⊗S C

εC⊗S idC��

C∆Coo ∆C // C ⊗S C

idC⊗SεC��

S ⊗S CϑlC

;;

C ⊗S SϑrC

cc

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 66: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Examples

Example

For any set X , the free S-semimodule S (X ) is anS-semicoalgebra:

∆(x) = x ⊗S x and ε(x) = 1S for all x ∈ X .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 67: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Semibialgebras

DefinitionWith an S-semibialgebra, we mean a datum (H , µ, η,∆, ε),where (H , µ, η) is an S-semialgebra and (H ,∆, ε) is anS-semicoalgebra such that

∆ : H → H ⊗S H and ε : H → S

are morphisms of S-semialgebras, or equivalently

µ : H ⊗S H → H and η : S → H

are morphisms of S-semicoalgebras.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 68: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Doi-Koppinen semimodules

Let B be an S-semibialgebra. As in the classical case, one canconsider several examples of Doi-Koppinen semimodules.

Hopf Modules

With SBB we will denote the category whose objects areS-semimodules M satisfying the following properties:

(M , ρM) is a right B-semimodules;(M , %M) is a right B-semicomodule;%M(mb) =

∑m < 0 > b1 ⊗S m < 1 > b2 ∀m ∈

M and b ∈ B .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 69: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Doi-Koppinen semimodules

Let B be an S-semibialgebra. As in the classical case, one canconsider several examples of Doi-Koppinen semimodules.

Hopf Modules

With SBB we will denote the category whose objects areS-semimodules M satisfying the following properties:

(M , ρM) is a right B-semimodules;(M , %M) is a right B-semicomodule;%M(mb) =

∑m < 0 > b1 ⊗S m < 1 > b2 ∀m ∈

M and b ∈ B .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 70: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Doi-Koppinen semimodules

Let B be an S-semibialgebra. As in the classical case, one canconsider several examples of Doi-Koppinen semimodules.

Hopf Modules

With SBB we will denote the category whose objects areS-semimodules M satisfying the following properties:

(M , ρM) is a right B-semimodules;

(M , %M) is a right B-semicomodule;%M(mb) =

∑m < 0 > b1 ⊗S m < 1 > b2 ∀m ∈

M and b ∈ B .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 71: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Doi-Koppinen semimodules

Let B be an S-semibialgebra. As in the classical case, one canconsider several examples of Doi-Koppinen semimodules.

Hopf Modules

With SBB we will denote the category whose objects areS-semimodules M satisfying the following properties:

(M , ρM) is a right B-semimodules;(M , %M) is a right B-semicomodule;

%M(mb) =∑

m < 0 > b1 ⊗S m < 1 > b2 ∀m ∈M and b ∈ B .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 72: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Doi-Koppinen semimodules

Let B be an S-semibialgebra. As in the classical case, one canconsider several examples of Doi-Koppinen semimodules.

Hopf Modules

With SBB we will denote the category whose objects areS-semimodules M satisfying the following properties:

(M , ρM) is a right B-semimodules;(M , %M) is a right B-semicomodule;%M(mb) =

∑m < 0 > b1 ⊗S m < 1 > b2 ∀m ∈

M and b ∈ B .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 73: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Automata ... Worthington 2009

DefinitionA left automaton (M ,A,�, s, γ) consists of:

1 an S-semialgebra A and a left A-semimodule (M ,�));2 an element s ∈ M, called start vector;3 an S-linear map γ : M → S.

A morphism of left automata

(ϕ, λ) : (M ,A,�, s, γ)→ (M ′,A′,�′, s ′, γ′),m 7→ ϕ(m)

is a pair consisting of a morphism of S-semialgebrasλ : A→ A′ and an A-linear map ϕ : M → M ′, s.t.:

ϕ(s) = s ′ and γ(m) = γ(ϕ(m)) for all m ∈ M .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 74: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Automata ... Worthington 2009

DefinitionA left automaton (M ,A,�, s, γ) consists of:

1 an S-semialgebra A and a left A-semimodule (M ,�));2 an element s ∈ M, called start vector;3 an S-linear map γ : M → S.

A morphism of left automata

(ϕ, λ) : (M ,A,�, s, γ)→ (M ′,A′,�′, s ′, γ′),m 7→ ϕ(m)

is a pair consisting of a morphism of S-semialgebrasλ : A→ A′ and an A-linear map ϕ : M → M ′, s.t.:

ϕ(s) = s ′ and γ(m) = γ(ϕ(m)) for all m ∈ M .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 75: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Application ... Worthington 2009DefinitionThe language accepted by the left automaton (M ,A,�, s, γ)is the S-linear map

ρM : A→ S , a 7→ γ(a � s) for all a ∈ A.

Example

Let B be an S-semibialgebra and (M ,B ,�, s, γ),(M ′,B ,�′, s ′, γ′) be left automata. Then(M ⊗bS M ′,B ,�M⊗bSM′ , s ⊗S s ′, γ ⊗S γ′) is a left automata.

b �M⊗SM′ m ⊗S m′ :=∑

b1m ⊗S b2m′.

Moreover, ρM⊗bSM′ = ρM ∗ ρM′ .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 76: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Application ... Worthington 2009DefinitionThe language accepted by the left automaton (M ,A,�, s, γ)is the S-linear map

ρM : A→ S , a 7→ γ(a � s) for all a ∈ A.

Example

Let B be an S-semibialgebra and (M ,B ,�, s, γ),(M ′,B ,�′, s ′, γ′) be left automata. Then(M ⊗bS M ′,B ,�M⊗bSM′ , s ⊗S s ′, γ ⊗S γ′) is a left automata.

b �M⊗SM′ m ⊗S m′ :=∑

b1m ⊗S b2m′.

Moreover, ρM⊗bSM′ = ρM ∗ ρM′ .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 77: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Application ... Worthington 2009DefinitionThe language accepted by the left automaton (M ,A,�, s, γ)is the S-linear map

ρM : A→ S , a 7→ γ(a � s) for all a ∈ A.

Example

Let B be an S-semibialgebra and (M ,B ,�, s, γ),(M ′,B ,�′, s ′, γ′) be left automata. Then(M ⊗bS M ′,B ,�M⊗bSM′ , s ⊗S s ′, γ ⊗S γ′) is a left automata.

b �M⊗SM′ m ⊗S m′ :=∑

b1m ⊗S b2m′.

Moreover, ρM⊗bSM′ = ρM ∗ ρM′ .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 78: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Application ... Worthington 2009DefinitionThe language accepted by the left automaton (M ,A,�, s, γ)is the S-linear map

ρM : A→ S , a 7→ γ(a � s) for all a ∈ A.

Example

Let B be an S-semibialgebra and (M ,B ,�, s, γ),(M ′,B ,�′, s ′, γ′) be left automata. Then(M ⊗bS M ′,B ,�M⊗bSM′ , s ⊗S s ′, γ ⊗S γ′) is a left automata.

b �M⊗SM′ m ⊗S m′ :=∑

b1m ⊗S b2m′.

Moreover, ρM⊗bSM′ = ρM ∗ ρM′ .Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 79: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Classical Examples

Example

Let (G , ·, e) be a monoid. Then S [G ] is an S-semibialgebrawith:

∆ : S [G ] → S [G ]⊗S S [G ], g 7→ g ⊗S g ;ε : S [G ] → S , g 7→ 1S .

Example

(S [x ],∆1, ε1) is an S-semibialgebra with:

∆1(xn) := xn ⊗S xn and ε1(xn) = 1S .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 80: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Classical Examples

Example

Let (G , ·, e) be a monoid. Then S [G ] is an S-semibialgebrawith:

∆ : S [G ] → S [G ]⊗S S [G ], g 7→ g ⊗S g ;ε : S [G ] → S , g 7→ 1S .

Example

(S [x ],∆1, ε1) is an S-semibialgebra with:

∆1(xn) := xn ⊗S xn and ε1(xn) = 1S .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 81: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Hopf Semialgebras

DefinitionA Hopf S-semialgebra is an S-semibialgebra H , along with anS-linear morphism S : H → H (called the antipode of H),such that∑

S(h1)h2 = ε(h)1H =∑

h1S(h2) for all h ∈ H .

Example

Consider the commutative semialgebra B := {0, 1} with1 + 1 = 1. Then B is a Hopf semialgebra:

∆(0) = 0⊗ 0, ∆(1) = 1⊗ 1ε(0) = 0, ε(1) = 1S(0) = 0, S(1) = 1.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 82: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Hopf Semialgebras

DefinitionA Hopf S-semialgebra is an S-semibialgebra H , along with anS-linear morphism S : H → H (called the antipode of H),such that∑

S(h1)h2 = ε(h)1H =∑

h1S(h2) for all h ∈ H .

Example

Consider the commutative semialgebra B := {0, 1} with1 + 1 = 1. Then B is a Hopf semialgebra:

∆(0) = 0⊗ 0, ∆(1) = 1⊗ 1ε(0) = 0, ε(1) = 1S(0) = 0, S(1) = 1.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 83: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Classical ExamplesExample

Let (G , ·, e) be a group. Then S [G ] is a Hopf S-semialgebra:

∆ : S [G ] → S [G ]⊗S S [G ], g 7→ g ⊗S g ;ε : S [G ] → S , g 7→ 1S .S : S [G ] → S [G ], g 7→ g−1.

Example

(S [x ], µ, η,∆2, ε2, S) is a Hopf S-semialgebra:

∆2(xn) :=n∑k=0

(nk

)xk ⊗S xn−k , ε2(xn) = δn,0

S : H → H , S(xn) = (−1)nxn.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 84: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Classical ExamplesExample

Let (G , ·, e) be a group. Then S [G ] is a Hopf S-semialgebra:

∆ : S [G ] → S [G ]⊗S S [G ], g 7→ g ⊗S g ;ε : S [G ] → S , g 7→ 1S .S : S [G ] → S [G ], g 7→ g−1.

Example

(S [x ], µ, η,∆2, ε2, S) is a Hopf S-semialgebra:

∆2(xn) :=n∑k=0

(nk

)xk ⊗S xn−k , ε2(xn) = δn,0

S : H → H , S(xn) = (−1)nxn.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 85: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Classical Examples ... continued

Example

(S [x , x−1], µ, η,∆, ε, S) is a Hopf S-semialgebra:

∆(xz) = xz ⊗S xz ;ε(xz) = 1S ;S(xz) = x−z .

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 86: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Fundamental Theorem of Hopf Semialgebras

TheoremConsider a S-semibialgebra B and the corresponding categoryof Hopf modules SBB . The following are equivalent:

1 B is a Hopf S-semialgebra;

2 B ⊗S B ' B ⊗bS B in SBB ;

3 B ⊗S B ' B ⊗cS B in SBB ;

4 HomBB (B ,−) : SBB → SS is an equivalence (with inverse−⊗S B).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 87: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Fundamental Theorem of Hopf Semialgebras

TheoremConsider a S-semibialgebra B and the corresponding categoryof Hopf modules SBB . The following are equivalent:

1 B is a Hopf S-semialgebra;

2 B ⊗S B ' B ⊗bS B in SBB ;

3 B ⊗S B ' B ⊗cS B in SBB ;

4 HomBB (B ,−) : SBB → SS is an equivalence (with inverse−⊗S B).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 88: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Fundamental Theorem of Hopf Semialgebras

TheoremConsider a S-semibialgebra B and the corresponding categoryof Hopf modules SBB . The following are equivalent:

1 B is a Hopf S-semialgebra;

2 B ⊗S B ' B ⊗bS B in SBB ;

3 B ⊗S B ' B ⊗cS B in SBB ;

4 HomBB (B ,−) : SBB → SS is an equivalence (with inverse−⊗S B).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 89: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Fundamental Theorem of Hopf Semialgebras

TheoremConsider a S-semibialgebra B and the corresponding categoryof Hopf modules SBB . The following are equivalent:

1 B is a Hopf S-semialgebra;

2 B ⊗S B ' B ⊗bS B in SBB ;

3 B ⊗S B ' B ⊗cS B in SBB ;

4 HomBB (B ,−) : SBB → SS is an equivalence (with inverse−⊗S B).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 90: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

J. Abuhlail, Semicorings and Semimodules, Talk, Brussel’sConference (2008).

E. Abe, Hopf algebras, Cambridge University Press (1980).

H. Al-Thani, Flat semimodules, Int. J. Math. Math. Sci.,2004(17) (2004), 873-880.

S. Caenepeel, G. Militaru and S. Zhu, Frobenius andseparable functors for generalized module categories andnonlinear equations, Springer-Verlag (2002).

S. Dascalescu, C. Nastasescu and A. Raianu, HopfSlgebras: an Introduction, Pure and Spplied Mathematics235, Marcel Dekker, New York (2001).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 91: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

K. Głazek, A Guide to the Literature on Semirings andtheir Spplications in Mathematics and InformationSciences. With Complete Bibliography, Kluwer ScademicPublishers, Dordrecht (2002).

J. Golan, Semirings and Their Spplications, KluwerScademic Publishers, Dordrecht (1999).

J. Golan, Power Algebras over Semirings. WithSpplications in Mathematics and Computer Science.Kluwer Scademic Publishers, Dordrecht (1999).

J. Golan, Semirings and Affine Equations over Them.Kluwer, Dordrecht (2003).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 92: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

J. Gómez-Torrecillas, Coalgebras and comodules over acommutative ring, Rom. J. Pure Sppl. Math. 43, 591-603(1998).

U. Hebisch and H. J. Weinert, Semirings: algebraic theoryand applications in computer science, World ScientificPublishing Co., Inc., River Edge, NJ (1998).

Y. Katsov, Tensor products and injective envelopes ofsemimodules over additively regular semirings, SlgebraColloq. 4(2) (1997), 121-131.

Y. Katsov, On flat semimodules over semirings. SlgebraUniversalis 51(2-3) (2004), 287-299.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 93: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

Y. Katsov, Toward homological characterization ofsemirings: Serre’s conjecture and Bass’s perfectness in asemiring context, Slgebra Universalis 52(2-3) (2004),197-214.

W. Kuich and S. Salomaa, Semirings, Automata,Languages, Springer-Verlag, Berlin (1986).

W. Kuich, Semirings, automata, languages,Springer-Verlag (1986).

S. Montgomery, Hopf Algebras and their Actions on Rings,SMS (1993).

H. Schubert, Categories, Springer-Verlag (1972).

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011

Page 94: Hopf semialgebras · Hopf semialgebras Jawad Y. Abuhlail (King Fahd University of Petroleum & Minerals, Saudi Arabia) abuhlail@kfupm.edu.sa In this talk, we introduce and investigate

PreliminariesThe category SS

Hopf Semialgebras

M. Sweedler, Hopf Algebras, New York: Benjamin, (1969).

M. Takahashi, On the bordism categories. II. Elementaryproperties of semimodules. Math. Sem. Notes Kobe Univ.9(2) (1981), 495-530.

M. Takahashi, On the bordism categories. III. FunctorsHom and for semimodules. Math. Sem. Notes Kobe Univ.10(2) (1982), 551-562.

R. Wisbauer, Foundations of Module and Ring Theory. SHandbook for Study and Research, Gordon and BreachScience Publishers (1991).

J, Worthington, Automata, Representations, and Proofs,Cornell University, August 2009.

Jawad Abuhlail Hopf Semialgebras Hopf algebras and tensor categories University of Almerıa (Spain) July 4 – 8, 2011