Unit – IV

Algebraic Structures Algebraic systems Examples and General Properties Semi groups and monoids Groups Sub groups Homomorphism Isomorphism

Binary and n-ary operations n-ary operation on nonempty set S Function from S X S X S X … X S to S (f : S X S X S X … X S S). Assigns a unique element of S to every ordered n-tuple of

elements of S. n order of the operation.

Unary operation on nonempty set S Assigns a unique element of S to every element of S. n-ary operation of order 1.

Binary operation on nonempty set S (*) Function from S X S to S (f : S X S S). Assigns a unique element of S to every ordered pair of elements (a,

b) of S. n-ary operation of order 2. a * b S is closed under the binary operation *.

Examples

Set of all integers (Z) is closed under addition(+), subtraction(–) and multiplication (*) operations.

Set of all real numbers (R) is closed under addition(+),

subtraction(–) and multiplication(*) operations.

Properties of Binary Operations

Let * and be binary operations on nonempty set S.

Commutative If a * b = b * a, for every a, b S.

Associative If a * (b * c) = (a * b) * c, for every a, b, c S.

Idempotent If a * a = a, for all a S.

Distributive a * (b c) = (a * b) (a * c) (a b) * c) = (a * c) (b * c), for all a, b, c S.

Examples

Addition and multiplication operations are commutative and associative on Z.

a + b = b + a, a + (b + c) = (a + b) + c a x b = b x a, a x (b x c) = (a x b) x c

Subtraction operation is neither commutative nor associative on Z.

a – b b – a, a – (b – c) (a – b) – c

Multiplication operation is distributive over the addition operation on Z.

a x (b + c) = (a x b) + (a x c) (a + b) x c = (a x c) + (b x c)

Addition operation is not distributive over the multiplication operation on Z.

a + (b x c) (a + b) x (a + c) (a x b) + c (a + c) x (b + c)

Let the binary operation * is defined on the set S = {a, b, c, d} as given in the operation table.

Element a * b is displayed in the (i, j) position. b * c = b c * b = d Operation * is not commutative.

b * (c * d) = b * b = a (b * c) * d = b * d = c Operation * is not associative.

* a b c d

a a c b d

b d a b c

c c d a b

d d b a c

Algebraic Systems – Examples and general properties

Algebraic system / Algebra / Algebraic Structure Some n-ary operations on nonempty set S. <S, *1, *2, …, *k>

Examples: <Z, +, x> <P(S), , >

Identity (e) Let * be a binary operation on nonempty set S.

el * x = x * er = x for every x in S.

Left Identity (e1) el * x = x for every x in S.

Right Identity (er) x * er = x for every x in S.

Inverse (x)

Let * be a binary operation on nonempty set S.

xl * a = a * xr = e.

a is invertible.

Left Inverse (x1)

Let * be a binary operation on nonempty set S.

xl * a = e.

a is left-invertible.

Right Inverse (xr)

Let * be a binary operation on nonempty set S.

a * xr = e.

a is right-invertible.

Standard Algebraic Structures

Ring

Let <R, +, .> be an algebraic structure for a nonempty set R and two binary operations + and . defined on it.

1) The operation + is commutative and associative.

a + b = b + a, for all a, b R.

a + (b + c) = (a + b) + c, for all a, b, c R.

2) There exists the identity element 0 in R w.r.t. +.

a + 0 = 0 + a = a, for every a R.

3) Every element in R is invertible w.r.t. +.

With every a R there exists in R its inverse element,

denoted by (–a).

a + (–a) = (–a) + a = 0.

4) The operation . is associative. a . ( b. c) = (a . b) . c for all a, b, c R.

5) The operation . is distributive over the operation + in R.

a . (b + c) = (a . b) + (a . c) (a + b) . c = (a . c) + (b . c) for all a, b, c R.

Zero element of the ring Identity element w.r.t. + the operation + (0).

Negative of a Inverse (–a) w.r.t. + of a R.

Examples 1. <Z, +, x>, Z is a set of integers and binary operations + and x.

2. <Q, +, x>, Q is a set of rational nos. and binary operations + and x.

3. <R, +, x>, R is a set of real nos. and binary operations + and x.

4. <C, +, x>, C is a set of complex nos. and binary operations + and x.

Commutative Ring If the operations +, . are commutative in a ring <R, +, .>.

Examples 1. <Z, +, x>, Z is a set of integers and binary operations + and x.

2. <Q, +, x>, Q is a set of rational nos. and binary operations + and x.

3. <R, +, x>, R is a set of real nos. and binary operations + and x.

4. <C, +, x>, C is a set of complex nos. and binary operations + and x.

Ring with Unity If the operations +, . have identity elements in a ring <R, +, .>.

Examples

1. <Z, +, x>, Z is a set of integers and binary operations + and x.

2. <Q, +, x>, Q is a set of rational nos. and binary operations + and x.

3. <R, +, x>, R is a set of real nos. and binary operations + and x.

4. <C, +, x>, C is a set of complex nos. and binary operations + and x.

Integral Domain

a . b = 0 a = 0 or b = 0 for a commutative ring with unity <R, +, .>.

Examples

1. <Z, +, x>, Z is a set of integers and binary operations + and x.

2. <Q, +, x>, Q is a set of rational nos. and binary operations + and x.

3. <R, +, x>, R is a set of real nos. and binary operations + and x.

4. <C, +, x>, C is a set of complex nos. and binary operations + and x.

Field If a ring <R, +, .>

is commutativehas the unityevery nonzero element of R has the inverse under the . operation.

Commutative ring with unity in which every nonzero element has a multiplicative inverse.

Examples 1. <Q, +, x>, Q is a set of rational nos. and binary operations + and x.

2. <R, +, x>, R is a set of real nos. and binary operations + and x.

3. <C, +, x>, C is a set of complex nos. and binary operations + and x.

4. <Z, +, x>, Z is a set of integers and binary operations + and x is not a field as Z does not contain multiplicative inverses of all its nonzero elements.

Exercises 1) Let S = {0, 1} and the operations + and . on s be defined by the

following tables:

Show that <S, +, .> is a commutative ring with unity.

+ 0 1

0 0 1

1 1 0

. 0 1

0 0 0

1 0 1

2) Let S = {a, b, c, d} and the operations + and . on s be defined by the following tables:

Show that <S, +, .> is a ring.

+ a b c d

a a b c d

b b a d c

c c d b a

d d c a b

. a b c d

a a a a a

b a a b a

c a b c d

d a a d a

Semigroups and Monoids

Semigroups

An algebraic system <S, *> consisting of a nonempty set S and an associative binary operation * defined on S.

Examples

1. <Z, +>, Z is a set of integers and binary operation +.

2. <Z, x>, Z is a set of integers and binary operation x.

3. <Z+, +>, Z+ is a set of positive integers and binary operation +.

4. <Z, –>, Z is a set of integers and binary operation – is not a semigroup.

Commutative / Abelian Semigroups

An algebraic system <S, *> consisting of a nonempty set S and an associative and a commutative binary operations * defined on S.

Examples

1. <Z, +>, Z is a set of integers and binary operation +.

2. <Z, x>, Z is a set of integers and binary operation x.

3. <Z+, +>, Z+ is a set of positive integers and binary operation +.

Monoid A semigroup with the identity element e w.r.t. *.

Examples 1. <Z, +> with the identity element 0.

2. <Z, x> with the identity element 1.

3. <P(S), > with the identity element .

4. <P(S), > with the identity element S.

Exercises Consider the binary operation * on a set A = {a, b} is defined

through a multiplication table. Determine whether <A, *> is a semigroup or a monoid or neither.

* a b

a b a

b a b

* a b

a a b

b a a

* a b

a a a

b b b

Consider the binary operation * on a set A = {a, b} is defined through a multiplication table. Determine whether <A, *> is a semigroup or a monoid or neither.

* a b

a a b

b b b

* a b

a a b

b b a

* a b

a b b

b a a

In each of the following, indicate whether the given set forms a semigroup or a monoid under the given operation.

1. The set of all positive integers, with a * b = maximum of a and b.

2. The set of all even integers on which the operation * is defined by a * b = ab / 2.

3. The set A = {1, 2, 3, 6, 9, 18} on which the operation * is defined by a * b = LCM of a and b.

4. The set Q of all rational nos. on which the operation * is defined by a * b = a – b + ab.

5. The product set Q x q, where Q is the set of all rational nos. on which the operation * is defined by (a, b) * (c, d) = (ac, ad + b).

Groups and Subgroups

Group (G)

Let a nonempty set G be closed under *.

Algebraic system <G, *> with the following conditions:

1. (a * b) * c = a * (b * c) for all a, b, c G (Associative).

2. There is an element e G such that a * e = e * a = a, for all a G (G contains identity element e under *).

3. For every a G, there is an element a’ G such that a * a’ = a’ * a = e (Every element a of G is invertible under * with a’ as an inverse).

Every group is a monoid and therefore a semigroup.

a2 = a * a ab = a * b

Abelian / Commutative Group If ab = ba for all a, b G.

Infinite Group A group G on a infinite set G.

Examples

1. <Z, +>

Associative. Identity element 0. Inverse is –a.

Infinite group.Abelian group (a + b = b + a. for all a, b Z).

2. Set of all non-zero rational or real or complex nos.

under multiplication.

Identity element 1.

Inverse is 1/a.

Infinite abelian group.

3. Set of all n x n non-singular matrices under matrix

multiplication.

Identity element is unit matrix of order n.

Infinite group.

Not abelian

(matrix multiplication is not commutative).

Subgroups

Let <G, *> be a group and H be a nonempty subset of G. Then <H, *> is a subgroup of G if <H, *> itself is a group.

Examples

1. The set of all even integers forms a subgroup of the group of all integers under usual addition.

2. The set of all nonzero rational nos. forms a subgroup of the group of all nonzero real nos. under usual multiplication.

Group Homomorphism and Isomorphism

Let G1 and G2 be two groups and f be a function from G1 to G2. The f is called a homomorphism from G1 to G2 if f(ab) = f(a)f(b), for all a, b G1.

The function f : G1 G2 is called an isomorphism from G1 onto G2 if

a. f is a homomorphism from G1 to G2.

b. f is one-to-one and onto.

The groups G1 and G2 are said to be isomorphic if there is an isomorphism from G1 onto G2.

Example Consider the groups <R, +> and <R+, x>.

Define the function f : R R+ by f(x) = ex for all x R.

Then, for all a, b R,

We have f(a + b) = ea+b = eaeb = f(a)f(b).

Hence f is homomorphism.

Take any c R+.

Then log c R and f(log c) = elog c = c.

Every element in R+ has a preimage in R under f.

f is onto.

For any a, b R,

f(a) = f(b)

ea = eb

a = b.

f is one-to-one.

f is an isomorphism.

Cosets and Lagrange's theorem

Let <G,*> be a group and <H,*> be a subgroup. For any a G,

let a*H = { a*h/ h H}

and H*a = {h*a / h H}.

Then, a*H is called the left coset of H w.r.t a in G and H*a is called the right coset of H w.r.t a in G

1. the left and right cosets of H are subsets of G

2. with each a G, there exists a left coset a*H of H and a right coset H*a of H . Further a = a*e a*H and a = e*a H*a.

3. the left and right cosets of H are not one and the same, in general.

4. If G is abelian, then every left coset of H is a right coset also.

5. e*H =H*e = H whenever or not G is abelian

Cosets and Lagrange's theorem

If the operation * is the addition +, we write a * H as a + H and H * a as H + a.

For example, consider the additive group of integers <Z,+> and its subgroup of even integers<E,+>. Then for any a Z,

The left coset of E w.r.t .a is a + E ={a + h / h E}

= { a +- 2n/ n z+}

= { a, a+-2,a+-4,a+-6,…….} And the right coset of E w.r.t. a is

E + a = {h + a /h E}

= {+-2n+ a/ n z+}

= { a, a+-2,a+-4,a+-6,…….}

Cosets and Lagrange's theorem

If G is a finite group and H is a subgroup of G, then the order of H divides the order of G