Cardinality and Algebraic Structures Dr Tijl De Bie Dept. Eng. Maths. email: [email protected].

Cardinality and Cardinality and Algebraic StructuresAlgebraic Structures

Dr Tijl De Bie

Dept. Eng. Maths.

email:

[email protected]

ContentsContents

Part I (weeks 1-7)• 1 Introduction• 2 Combinatorics, permutations and

combinations.• 3 Algebraic Structures and matrices:

Homomorphism, isomorphism, group, semigroup, monoid, rings, fields

• 4 Lattices and Boolean algebras

• If time remains: some illustrations of the use of group theory in cryptography

Part II (weeks 8-12)• Vector spaces

IntroductionIntroduction• Computer programs frequently handle real world

data.

• This data might be financial e.g. processing the accounts of a company.

• It may be engineering data e.g. from sensors or actuators in a robotic system.

• It may be scientific data e.g. weather data or geological data concerning rock strata.

• In all these cases data typically consists of a set of discrete elements.

• Furthermore there may exist orderings or relationships among elements or objects.

• It may be meaningful to combine objects in some way using operators.

• We hope to clarify our concepts of orderings and relationships among elements or objects

• We look at the idea of formal structures such as groups, rings and and formal systems such as lattices and Boolean algebras

Number SystemsNumber Systems

• The set of natural numbers is the infinite set of the positive integers. It is denoted N and can have different representations:

{1,2,3,4,........}

{1,10,11,100,101,.....}

are alternative representations of the same set expressed in different bases. Nm is the set of the first m positive numbers i.e. {1,2,3,4, ......,m}. N0 is the set of natural numbers including 0 i.e. {0,1,2,3,5,....}

• Q denotes the set of rational numbers i.e. signed integers and fractions

{0,1,-1,2,-2,3,-3,....,1/2,-1/2,3/2,-3/2,5/2,

-5/2,....,1/3,-1/3,2/3,-2/3,........}

• R is the set of real numbers i.e. the coordinates of all the points on a line.

• Z is the set of all integers, both positive and negative {0,1,-1,2,-2,3,-3,......}

2 Combinatorics: Permutations2 Combinatorics: Permutations

• A permutation of the elements of a set A is a bijection from A onto itself.

• If A is finite we can calculate the number of different permutations. Suppose A={a1,...,an}

a1

nchoices

n-1choices

1choice

a2 an

total number of ways of filling the n boxesn x (n-1)x(n-2)x(n-3)..............x1=n!

nPn=n!

eg a possible permutation of {1,2,3,4,5,6} is

1 2 3 4 5 6

5 6 3 1 4 2

Composition of PermutationsComposition of Permutations

• If :A A and :A A are permutations of A then the composition or product .of and satisfies for all x in A

.x)= (x))

Notice that since both and are bijections from A into A so is . In other words . is a permutation of A.

• Example: Let A={1,2,3,4,5,6} then two possible permutations are

1 2 3 4 5 6

5 6 3 1 4 2

1 2 3 4 5 6

3 2 6 1 4 5

For . we have that

1 5 4,2 6 5,3 3 6

4 1 3,5 4 1,6 2 2

. 1 2 3 4 5 6

4 5 6 3 1 2

Cyclic PermutationsCyclic PermutationsA cyclic permutation on a set A of n elements has the form where :

a1 a2 ak-1 ak ak +1 an

a2 a3 a k a1 ak +1 an

k n

For shorthand we often write a1 a2 ak

Example

6 1 4 2 3 5

1 4 6 2 3 5

or (6 1 4) is a cyclic permutation

Two cyclic permutations a1 a2 ak and b1 b2 b t are said to be disjoint if

a1, ,ak b1 ,, bt

e.g. (4 5 2) and (3 1 6) are disjoint

is said to be a k cycle

Notice that

1 2 3 4 5 6

5 6 3 1 4 2

1 5 4 2 ,6 3

Other examples are

1 2 3 4 5

4 2 5 3 1

1 2 3 4 5

4 2 5 3 1

3 5 1 4 2

or

1 2 3 4 5 6

2 3 1 5 4 6

1 2 3 4 5 6

Can you spot a product of disjoint cyclic permutations equivalent to the following permutation ?

1 2 3 4 5 6 7

1 7 4 6 2 3 5

• Theorem: Every permutation of a finite set A can be expressed as a combination of disjoint cycles.

Structure underlying permutations

Note that the following hold:

(1) The product of two permutations is a uniquely determined permutation of the same set.

(2) The composition of permutations is associative.

(3) The permutation

I =

a1 a2 an

a1 a2 an

is called the identity permutation and has theproperty that I. = .I =

(4) For every permutation

=

a1 a2 an

b1 b2 bn

1 =

b1 b2 bn

a1 a2 an

there is an inverse

such that

. 1 1 . I

CombinationsCombinations• When we think about combinations we do not

allow repeats and unlike permutations we do not consider order.

• Combinations look at the number of different ways of picking a subset of k elements from a set of n elements.

• Think of the number of ways of picking a list of k distinct elements of n

n n-1 n-k-2 n-k-1no. of choices

places

= n(n-1)(n-2) ........... (n-k-1) = n!/(n-k)!

For each possible list there are k! permutationsso since we are not interested in order we should divide the above by k!.

C(n,k) = Cnk = n!/(n-k)!k!

• Example: Choosing 2 elements from {a,b,c,d}

{a,b},{a,c},{a,d},

{b,c},{b d},{c,d}

C(4,2)= 4!/(2! 2!) =6

Combinations with Repetitions

We could also consider combinations with repetitions. With repetitions the number of distinct combinations of k elements chosen from n is:

C(n+k-1,k)= (n+k-1)!/k!(n-1)!

Number of different throws of 2 identical dice

(1 1)(2 2)(3 3)(4 4)(5 5)(6 6)

(1 2)(1 3)(1 4)(1 5)(1 6)

(2 3)(2 4)(2 5)(2 6)

(3 4)(3 5)(3 6)(4 5)(4 6)(5 6)

C(7,2)=21

Algebraic StructuresAlgebraic Structures

• When we consider the behaviour of permutations under the composition operation we noticed certain underlying structures.

• Permutations are closed under this operation, they exhibit associativity, an identity element exists and an inverse exists for each permutation

• These properties define a general type of algebraic structure called a group.

• In this section we shall look at groups in more detail as well as other similar algebraic structures such as semigroups and monoids.

• Later we will progress to consider more complex algebraic structures such as rings, integral domains and fields.

• We will see that many real life situations are examples of these algebraic structures

GroupsGroupsA group G, or G, is a set G with binary

operation which satisfies the following properties

1. is a closed operation i.e. if a G and

b G then a bG

2. a,b,c G a b c a b c this is theassociative law

3. G has an element e, called the identity, such that a G a e = e a = a

4. a G there corresponds an element

a -1 G such that a a-1 a-1 a = e

Example:

The set of all permutations of a set Aonto itself is group (called the symmetric group Sn for n elements).

Group of Symmetries of a Group of Symmetries of a TriangleTriangle

Consider the triangle

X

Y Z

O

l

nm

We can perform the following transformationson the triangle1=identity mapping from the plane to itselfp=rotation anticlockwise about O through 120 degreesq=rotation clockwise about O through 120 degreesa=reflection in lb=reflection in mc=reflection in n

Let x y denote transformation y followed by

transformation x for x and y in {1,p,q,a,b,c}

So for example p a = c

l

m Y n

O

X

Z

l

m n

O

l

m n

O

X

YZ X Z

Ya p

1 p q a b c

1 1 p q a b c

p p q 1 c a b

q q 1 p b c a

a a b c 1 p q

b b c a q 1 p

c c a b p q 1

Notice the table is not symmetric

Other examples of a groupOther examples of a group

• The set of all permutations onto itself is a group (called the symmetric group Sn)

• The sets of all invertible nxn matrices forms a group under ordinary matrix multiplication (called GL(n), the general linear group)

• The quaternion group: Let G={I,-I,J,-J,K,-K,L,-L} where

I=[ ], J=[ ], K=[ ] , L=[ ]1 00 1

j 00 -j

0 1-1 0

0 jj 0

Order of a groupOrder of a group

• A finite group is a group where G is finite

• The order of a finite group is |G|

• For example if G is the set of permutations of a set A with n elements then the order of G is n!

Abelian GroupsAbelian Groups

If G, is a group and is also commutative

then G, is referred to as an Abelian group

(the name is taken from the 19’th century mathematician N.H. Abel)

is commutative means that

a,b G, a b = b a

Examples: R,+ , Z , and R - 0 , are abelian groups.

Why is R, not a group at all?

Modular arithmetic

• Recall a=b mod p iff p|a-b

• Notice a=b mod p iff a=kp+b for some integer k

a=b mod p implies p|a-b impliesa-b=kp implies a=kp+b

a=kp+b implies a-b=kp impliesp|a-b implies a=b mod p

Modular addition

• Modular addition mod 6:

+ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 1 2 3 4 5 0

2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

Modular multiplication

• Modular multiplication mod 7:

x 1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 4 6 1 3 5

3 3 6 2 5 1 4

4 4 1 5 2 6 3

5 5 3 1 6 4 2

6 6 5 4 3 2 1

Modular multiplication

• Modular multiplication mod 6:

x 1 2 3 4 5

1 1 2 3 4 5

2 2 4 0 2 4

3 3 0 3 0 3

4 4 2 0 4 2

5 5 4 3 2 1

Modular multiplication

• Not a group! (Why not?)• Which subset of {1,2,3,4,5} does form a group?

x 1 2 3 4 5

1 1 2 3 4 5

2 2 4 0 2 4

3 3 0 3 0 3

4 4 2 0 4 2

5 5 4 3 2 1

Modular multiplication

Theorem: If n>=2 and n|p then n has no inverse under multiplication mod p

Prove it!

The subset of {1,…,p-1} relatively prime to p is a group under multiplication mod p denoted Zp

*

We will clarify this on the next slides…

Modular arithmetic

• Recall Euclid’s algorithm to find the gcd of x and y:

x=k1y+r1

y=k2r1+r2

r1=k3r2+r3

…

rn-2 =kn-1rn-1+rn

rn-1=knrn

From this… Theorem:There exist integer a and b such

ax+by=gcd(x,y)

The old remainder is divided by the new one repeatedly until the remainder is 0The gcd is the last non zero remainder

Modular arithmetic

• An element n has an inverse n-1 under multiplication mod p for which

n. n-1 =1 mod p

if and only if (iff) n is relatively prime to p.

• Prove this!

• Clearly then if p is prime then every element will have an inverse.

Groups in logic

Consider exclusive or defined by

A B ⊕ ≡(¬A B) (A ¬B)∧ ∨ ∧

• {t,f} is an abelian group under exclusive or.

• What is the identity?

• What is the inverse of t (and f)?

Two show that an algebraic system is a group we must show that it satisfies all the axioms of a group.

Question: Let A, , , be a Boolean algebraso that A is a set of propositional elements, islike ‘or’, is like ‘and’ and is like ‘not’. Show

that A, is an abelian group where

a,b A a b = a b a b Answer:(1) Associative since ab c = a b c

prove this ?

(2) Has an identity element 0 (false) sincea a0 = a 0 a 0 a 1 0

a 0 = a

(3) Each element is its own inverseaa = a a a a 0 0 0

(4) The operation commutes a b = b aprove this ?

Iterated operations

• a=a1

• a◦a=a1

• a◦a◦a=a2

• a◦a…◦a=ak

• (Why is this unambiguously defined?)

Cyclic groups

A group G is cyclic if there exists a G such ∈that for any b G there is an integer k≥0 ∈such that ak=b.

I.e. Every element of G is some power of a.Element a is called the generator of G

denoted G=<a>

Example:• <{1,-1},×>=<-1> since –12=1, -13=-1

Order of a cyclic permutation group

• (1 2 … p)

• Show that the order is equal to p

• [Show by making a drawing…]

Weaker structures

• An Abelian group is a strengthening of the notion of group (i.e. requires more axioms to be satisfied)

• We might also look at those algebraic structures corresponding to a weakening of the group axioms

Semigroup monoid group Abelian Group⊆ ⊆ ⊆

SemigroupSemigroup

A, is a semigroup if the following conditionsare satisfied:

1. is a closed operation i.e. if a A and

b G then a bA

2. is associative

Example: The set of positive even integers {2,4,6,.....} under the operation of ordinary additionsince• The sum of two even numbers is an even number• + is associative

The reals or integers are not semigroups under -why?

MonoidMonoidA, is a monoid if the following conditionsare satisfied:

1. is a closed operation i.e. if a A and b G thena bA

2. is associative3. There is an identity element

Examples: Let A be a finite set of heights. Let be a binary operation such that a bis equal to the taller of a and b. Then A,

is a monoid where the identity is the shortest person in A

true, false , is a monoid: is associative,

true is the identity, but false has no inverse

true, false , is a monoid: is associative

false is the identity, but true has no inverse

Properties of Algebraic Properties of Algebraic StructuresStructures

Semigroup monoid group Abelian Groupproperties

Theorem: (unique identity) Suppose that A,is a monoid then the identity element is unique

Proof: Suppose there exist two identity elementse and f. [We shall prove that e=f]

e = e f since f is an identity = f since e is an identity

Theorem: (unique inverse) Suppose that A,is a monoid and the element x in A has an inverse.Then this inverse is unique.

Proof: ??

Properties of GroupsProperties of GroupsTheorem (The cancellation laws): Let G , bea group then a,x,y G

(i) a x = a y x = y

(ii) x a = y a x = y

Proof: (i) Suppose that a x = a y then by axiom 3

a has an identity a-1

and we have that

a -1 a x a -1 a y a-1 a x = a -1 a y associativity e x = e y a-1 is the inverse x y identity (ii) is proved similarly

Theorem (The division laws): Let G , bea group then a,x,y G

(i) a x = b x = a -1 b

(ii) x a = b x = b -1 a

Proof ??

Theorem (double inverse) :If x is an element ofthe group thenG ,

x -1 -1 = x

Proof:

x -1 -1 x-1 = e x-1 -1 is inverse of x -1 x-1 -1 x-1 x = e x = x

x-1 -1 x-1 x x associativity

x-1 -1 e = x x -1 is inverse of x x-1 -1 = x identity

Theorem (reversal rule) If x and y are elements of the group thenG ,

x y 1 y-1 x -1

Proof ??

For an arbitrary element of a group G , wecan define functionsa : G G and a : G G

such thatxG a x a x and a x x a

Theorem: a : G G and a : G G

are permutations of G

Proof: Consider a

[prove 1-1] suppose for x,y in G

a x a y a x = a y x = y (cancellation laws)

[Prove onto] For any y in G

a a-1 y a a-1 y a a-1 y (associativity)

= e y (a-1 is inverse of a)

= y (identity)

Corollary: In every row or column of the multiplication table of G each element of G appearsexactly once.

SubgroupsSubgroupsH, is a subgroup of the group G, if H G

and H, is also a group

Examples: Q - 0 , is a subgroup of R - 0 ,1, 1, i,-i , is a subgroup of C - 0 ,

Test for a subgroupLet H be a subset of G. Then H, is a subgroupof G, iff the following conditions all hold:

(1) H

(2) H is closed under multiplication

(3) x H x -1 H

For every group G, , G, and e , aresubgroups

e , is called the trivial subgroup of G,

a proper subgroup of G, is a subgroup

different from G

A non-trivial proper subgroup is a subgroupequal neither to or to G, e ,

CosetsCosetsConsider a set A with a subset H. Let a A .Then the left coset of H with respect to a isthe set of elements:

a x x H

This is denoted by a H

Similarly the right coset of H with respect to a is

x a x H and is denoted by H a

Example: Let A be the set of rotations

0 ,60 ,120 ,180 ,240 ,300 and

H 60 ,120 ,240 . Let a = 60 then

x a x H 60 ,180 ,300 which is the right coset with respect to

60

{0º,120º,240º}

Normal SubgroupsNormal Subgroups

Let H, be a subgroup of G, . Then H,

is a normal subgroup if, for any a G , the left

coset a H is equal to the right coset H a

H, is a normal subgroup where H = , , e.g. H = , , , , H , , ,,

Theorem: In an Abelian group, every subgroupis a normal subgroup

Coset cardinality

Theorem: For any H subset of G and any a in G |a•H|=|H|

Proof:

By definition of Coset |a•H|≤|H|Now suppose |a•H|<|H| then there must exist

b and c distinct elements of H such that a•b=a•c.

But by the cancellation law this implies that b=c which is a contradiction.

Hence |a•H|=|H|

Coset partitioning

Theorem: Let a,b G and let H be a ∈subgroup of G then either:a•H=b•H or: a•H∩ b•H=∅

Proof:Suppose a•H∩ b•H≠ then there exist s and t in ∅

H such that a•s=b•t.In this case a= b•t•s-1 and for an arbitrary x in H

a•x= b•t•s-1•xNow by the inverse axiom and closure,

t•s-1•x H and hence b•t•s∈ -1•x b•H, therefore ∈a•x b•H so that a•H b•H∈ ⊆

Similarly we can show that b•H a•H⊆Hence if the two cosets are not disjoint then

b•H=a•H

LeGrange’s theorem

Theorem: Let H be a subgroup of finite group G, then the cardinality of H evenly divides the cardinality of G (i.e |H| | |G|)

Proof

Let |G|. Now for each element ai of G we can generate a coset ai•H.

Now notice that ai a∈ i•H because since H is a subgroup, e H and a∈ i•e= ai

Suppose there are m distinct cosets of H then picking one representative ai from each this means that:G= a1•H a∪ 2•H a∪ 3•H … a∪ m•H

LeGrange’s theorem

Now by the previous theorem it follows that since these m cosets are distinct then they must be disjoint.

Hence,

|G|=|a1•H|+ |a2•H| + |a3•H| … + |am•H|

Also by the cardinality theorem for cosets they all have the same cardinality, namely |H|. Hence, |G|=m.|H| as required

Order of an element

• Let i be the smallest integer such that ai=e where a is an element of group G and e is the identity element.

• If i exists we call it the order of a.

• Otherwise we say that a has infinite order.

Subgroup generated by an element

Theorem: For any element a of G with finite order the set:H={aj: for some integer j}is a subgroup of G.

Notice: if i is the order of element a then• ai=e• ai+1=e•a=a1

• ai+2= a •a =a2

• ai+n=an

Example

• Let σ=(1 2 3 4), a permutation of 4 elements

• Then {σ, σ2, σ3, σ4} is a subgroup of the group of permutations of {1,2,3,4}

• The order of σ is 4

• [Work it out!]

Order of elements in finite groups

• If the group G is finite then all elements of G have finite order:

• For any a G, since G is finite there ∈must exist i<j such that ai=aj

• a•ai-1=a•aj-1 cancellation law implies ai-1=aj-1

• Repeated application of the cancellation law gives a=aj-i+1

• a•e=a•aj-i implies e=aj-i

Corollary of LeGrange

Theorem: The order of every element of a finite group G, divides the order of G

Proof...Every element of G has finite order n

and hence generates a subgroup of order n.

Hence by LeGrange’s theorem n divides |G|

IsomorphismIsomorphism

• Two groups are isomorphic if there is a bijection of one onto the other which preserves the group operations i.e.

if G1 , and G2 , are groups then a bijectionf : G1 G2 is an isomorphism provided

x,y G1 f x y f x f y

Example: Consider the group of matricesof the form where under matrix

1 t

0 1

t R

multiplication. This is isomorphic to the group

The mapping is

R,+1 t

0 1

t

An isomorphism from a group onto itself iscalled an automorphism.

HomomorphismsHomomorphisms The idea of isomorphic algebraic structures

can be readily generalised by dropping the requirement that the functional mapping be a bijection.

Let A, and B, be two algebraic systemsthen a homomorphism from A, to B,

is a functional mapping f : A Bsuch that

x,y A f x y f x f y Example: consider the two structures

1 0 1

1 1 1 0

0 1 0 1

1 0 1 1

then f such that f =1, f = 1,f 1,f 0

f 0,f 1 is a homomorphism between

,, , ,, , and 1,0 , 1 ,

Algebraic Structures with two Algebraic Structures with two OperationsOperations

• So far we have studied algebraic systems with one binary operation. We now consider systems with two binary operations.

• In such a system a natural way in which two operations can be related is through the property of distributivity;

Let A,, be an algebraic system with twobinary operations and . Then the operation is said to distribute over the operation ifx,y, z A x y z x y x z and

y z x = y x z x

Example: distributes over +

distributes over distributes over

RingRing

An algebraic system A,, is called a ring if

the following conditions are satisfied:

(1) A, is an Abelian group(2) A, is a semigroup(3) The operation is distributive over the

operation

Example: Z, +, is a ring sinceZ, + is an Abelian group

Z, is a semigroup distributes over +

Examples of rings

<Z, +, ×> is a ring because:• <Z, +> is an Abelian group.• <Z, ×> is a semigroup.• × distributes over +

The set {[ ],a,b є R}is a ring under matrix addition and multiplication

{0,1,…,n-1} is a ring under addition and multiplication mod n

0 a0 b

Rings of polynomials

• Let the set R[x] be the set of all polynomial of the form:anxn+…+ a2x2+ a1x1+a0

for some n, where an,…,a0 єR

• Then R[x] is a ring under addition and multiplication of polynomials

• In fact for any ring R you can construct a ring of polynomials R[x] over R

Special types of ring

A commutative ring is a ring in which iscommutative

A ring with unity contains an element 1 suchthatxA x 1 = 1 x = x where 10

Example: the ring of 2x2 matrices under matrixaddition and multiplication is a ring with unity.The element 1=I=

1 0

0 1

(0 is the identity of )A,

Division rings

• A division ring is a (not necessarily commutative) ring with unity, in which every element a not equal to 0 has an inverse a-1 such that a•a-1= a-1•a=1

• The ring of complex matrices of the form:

[ ]a b-b a

Integral Domains and FieldsIntegral Domains and FieldsA,, is an integral domain if it is a commutativering with unity that also satisfies the followingproperty;

x,y A x y = 0 x = 0 or y = 0

Z, +, is also an integral domain

A,, is a field if:

(1) A, is an Abelian group

(2) A - 0 , is an Abelian group

(3) The operation is distributive over the operation

Example:The set of real numbers with respect to+ and is a field.

Z, +, is not a field. Why?

Galois fields

• For a prime number p the set {0,1,…,p-1} is a field under modular addition and multiplication mod p

• A field (like this one) with finite number of elements is called a Galois field.

A Field is an Integral Domain A Field is an Integral Domain Let A,, be a field then certainly A,,

is a commutative ring with unity. Hence, it onlyremains to prove that

x,y A x y = 0 x = 0 or y = 0

Now suppose x y = 0 then if x=0 the above holds. Consider the case then where x 0

Since A - 0 , is an Abelian group then it

must contain an inverse to x, x -1 , for which the

following holds

y = 1 y = x -1 x yx -1 x y x -1 0

Nowa 0 0 a 0

a 0a 0 = a 0 (distributivity)

a 0a 0 = a 0 0 0 is identity a 0 = 0 cancellation laws for

Therefore y=0 as required

Properties of a ringProperties of a ring

Theorem: if A,, is a ring. ThenxA 0 x = x 0 = 0

Proof: as for previous argument

Let -x denote the inverse of x under

Theorem: if A,, is a ring then the following

hold(i) -x y = x -y - x y (ii) -x -y x y

Proof: (i)

x -x y = 0 y (additive inverse)

0 (by above theorem)

x y -x y = 0 (distributivity)

-x y = - x y 0 (division laws for )

= - x y (additive identity)

(ii) -x -y x -y (part(i))

= - - x y (part(i))

= x y (double inverse)

for both (i) and (ii) the symmetric cases areproved similarly

Property of an integral domain

Theorem: suppose that elements a,b and c ofan integral domain satisfy anda b = a c a 0

then b=c.Proof:

a b - a c a c - a c 0 (additive inverse)

Now - a c a -c (prev. theorem)

a b c 0 (distributivity)

b c 0 by defn. of integer domain

since a 0

b = 0 - -c (by devision law for )

b = c double inverse

Subrings and subfield

Subring• If (A, ,•) is a ring then (H, ,•) is a ⊕ ⊕

subring if H A and⊆• (H, ,•) is a ring⊕

Subfield• If (A, ,•) is a field then (H, ,•) is a ⊕ ⊕

subfield if H A and⊆• (H, ,•) is a field⊕

Examples: Z is a subring of R, R is a subfield of C

Ring morphisms

A morphism between rings (A, ,•) and ⊕(B,*, ) is a function f:A→B such ⊗that: x,y A∀ ∈

• f(x y)=f(x)*f(y) and⊕• f (x•y)=f(x)•f(y)

From these we have that• f(0)=0′ where 0′ is the zero of

(B,*, )⊗• Also f(-x)=-f(x)

Special morphisms

1. An injective ring morphism is called a monomorphism

2. A surjective ring morphism is called an epimorphism

3. A bijective ring morphism is called a isomorphism

Examples of morphisms

• f(a) = a mod n, is an epimorphism (surjective ring morphism) between Z and {0,1,…,n-1}

• For the ring of polynomials R(x), f(p)=p(j) is an epimorphism into C, where p(j) is obtained by substituting j for x in the polynomial p

Galois theorem

• For every prime power pk (k=1,2,…) there is a unique (upto isomorphism) finite field containing pk elements denoted by GF(pk)

• All finite fields have cardinality pk

Galois theorem: examples

• GF(2)+ | 0 1 · | 0 1--+---- --+---- 0 | 0 1 0 | 0 0 1 | 1 0 1 | 0 1

• GF(3)+ | 0 1 2 · | 0 1 2 --+------ --+------ 0 | 0 1 2 0 | 0 0 0 1 | 1 2 0 1 | 0 1 2 2 | 2 0 1 2 | 0 2 1

Partial OrderingsPartial Orderings• We have introduced formal structure governingthe properties of various sets of elements under one or two binary operations. These elements can also be ordered and restricted by binary relations.• In this section we revise our understanding of binary relations in a set and also introduce a graphical notation for binary relations.

A relation R on a set A is a partial order if itsatisfies;(1) R is reflexive xA R(x, x)

(2) R is antisymmetric

y=xxy,R and yx,RA yx,

(3) R is transitivex,y, z A R x,y and R y, z R x,z

Example: Set of reals R with the relation

The pair (A,R) is called a partially ordered setor poset

Example: The relation can be defined on a Boolean algebra by;

x y iff x y = y ( is the logical or)

(1) Thus from the idempotent law x x = x

we find that x x and hence the relation is

reflexive.

(2) If x y and y x then x y = y and y x = x

From the commutative law x y = y x

and hence the relation is antisymmetric(3) If x y and y z then

x y = y and y z = z

z = x y z

x y z (associative law)

= x z

x z

We can think of a relation as being represented by the set of pairs of elements which satisfy the relation.In this case a partial ordering on A corresponds to a subset B of AxA satisfying

xA x,x B

x,y A x, y B and y, x B x = y

x,y, z A x,y B and y, z B x,z B

Other examples of partial orderings:Divisibility on N: We say that a divides b iff thereis some x in Z such that ax=b. If this divisibilityexists we write a|b. Divisibility is a partial order on N.

Inclusion on a set of sets X

Graphical RepresentationsGraphical Representations

We can represent partial orderings graphicallyby means of a directed graph where the nodesare elements of A and the directed edges givethe partial order relations. e.g. the graph

a

b c

d

Denotes the partial ordering on {a,b,c,d}

where

a a, bb, cc, d d

b a, ca, d a, d b

Graphical Representations of Graphical Representations of the Axiomsthe Axioms

Reflexive:

a

Antisymmetric: the following does not occur

a

b

Transitive:

a c

b

Example: Divisibility relation on{2, 3, 4, 6, 8, 9, 18}2|4 4|8 2|8 2|6 3|6 3|9 9|18 6|18 3|18 2|18

3 9 18

6

2 4 8

Example: The collection of all subsets of {a,b,c}

{a,b,c}

{a,b} {a,c} {b,c}

{a} {b} {c}

Hasse DiagramsHasse Diagrams

Notice that some of the diagrams in the previousexamples were messy and difficult to read havingmany links.

We can simplify these diagrams by introducingcertain conventions.

The Hasse diagram of a partially order set is a drawing of the points in the set (as nodes) andsome of the links of the graph of the order relation.

The rules for drawing the Hasse diagram of a partialorder are:(1) Omit all links that can be inferred from transitivity.(2) Omit all loops(3) Draw links without arrow heads(4) Understand that all arrows would point upwards

Here are Hasse diagrams for the two examplesgiven previously:

2

4

8

6

3

9

18

Divisibility:

{a,b} {a,c} {b,c}

{a} {b} {c}

Example: subsets

{a,b,c}

Incomparable ElementsIncomparable Elements

Consider the Hasse diagram for divisibility on{2,3,....,10}

2

4

8 10

6 9

3 5 7

Notice that 5 and 6 are not related in either directionSimilarly for 2 and 3

If neither R(a,b) or R(b,a) then a and b are incomparable or not comparable

Linear or Total OrderLinear or Total Order

A linear or total order on a set A is a partial orderon A in which every two elements are comparable

a,b A either R a, b or R b,a

2

3

4

5

Maximal and Minimal Maximal and Minimal ElementsElements

A maximal element of A is any element t ofA such that xA R(t, x) x = t

A minimal element of A is any element b of A such that xA R(x, b) x = b

Example:For the subset ordering {a,b,c} is themaximal element and is the minimal element

For divisibility on {2,.....,10} the maximal elements are 6, 7, 8, 9 and 10and the minimal elements are 2, 3, 5 and 7The element 4 is neither maximal nor minimal

Upper Bounds and Upper Bounds and Lower BoundsLower Bounds

Let S be a subset of A then x in A is an upperbound of S if yS R y, x

Similarly z in A is a lower bound of S ifyS R z, y

An element u is the least upper bound of S if u is an upper bound of S and for every x anupper bound of S R(u,x)

An element l is the greatest lower bound of S if l is an upper bound of S and for every z alower bound of S R(z,l)

The least upper bound (lub) of S is sometimes referred to as the supremum of S (sup S)

The greatest lower bound (glb) of F is sometimesreferred to as the infimum of S (inf S)

LatticesLattices

A partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound is called a lattice.

a b

c d

e f

This is not a lattice since {c,d} has no lub orglb.

A lattice in which every subset has a lub and glbis called complete.Every finite lattice is complete.

For a complete lattice the lub of the whole latticeis call top and the greatest lower bound bottom

Example: Consider elements of the form (a,b,c)where a,b and c can take the values 0 or 1. Fortwo such elements f and g we say that f gif each coefficient of f is less than or equal tothe corresponding coefficient of ge.g. 001 011 but not 001 010

(111)

(011) (101)(110)

(100)(010)(001)

(000)

Meet and JoinMeet and Join

In a lattice A, the following equations

define binary operations on A

x y = lub x,y x y = glb x,y

is called the meet operation and iscalled the join operation.

They have the following properties

a b = b aCommutativity:

Associativity: a b c = a b c

Since a b is an upper bound of a and b

a a b

Similarly for the meet

a b a

If a b and cd thena cb d

a cb d

Theorem

Proofb b d (by definition)

d b d (similarly)

a b and c d (given)

b d is an upper bound of a and c

a c is the least upper bound of a and c

a cb d

Let 1 denote the lub of the whole lattice and 0denote the glb of the whole lattice. Then

xL 0 x1

x 1 = 1 x 1 = x

x 0 = 0 x 0 = x

ExampleExampleLet us order the following set of numbers with the operation “is a factor of”. A={3,9,12,15,36,45}

45

9

36

12

3

15

The join operation is the least common multiple

The meet operation is the greatest common divisor

Complemented LatticeComplemented Lattice

For a complemented lattice we have that forxL there exists x L such that:

x x = 0 and x x = 1

e.g.

a b c

a b c

1

0

Distributive LatticeDistributive Lattice

A lattice is distributive if:

x y z x y x z x y z x y x z

e.g. the following lattice is not distributive

a

b c d

e

b d c b e = b

b d b c a a = a

Since

Boolean AlgebraBoolean AlgebraA Boolean Algebra consists of two binary operations and and the unary operation on a set B with distinct elements 0 and 1 such that the following hold.

(1) The commutative laws:

x y = y x

x y = y x

(2) The associative laws:

x y z = x y z x y z = x y z

(3) The Distributive laws:

x y z x y x z x y z x y x z

(4) The Identity Laws:

x 0 = x

x 1 = x

(5) The Complementation Laws:

x x =1

x x = 0

If L, is a complemented distributive lattice thenL, , , is a Boolean algebra where , and

correspond to the meet, join and complement operations on L respectively

Theorem

Proof ?

(6) The following Idempotent Laws can be derived:

x x = x

x x = x

Proofx x = x x 1 identity law (4) x x x x (complementation law (5))

x x x (distributive law (3))

= x 0 (complementation law (5))

= x

(7) The following Identity Laws can also be derived

x 1 = 1,x 0 = 0

Proof

x 1 = x x x (complementation law (5))

= x x x (associative law (2) )

= x x (idempotent law (6))

= 1 (complementation law (5))