Algebraic Structures - Definitions - Examples - Properties

ALGEBRAIC SYSTEMS - DEFINITIONS - EXAMPLES - PROPERTIES

Definition 1: Algebraic system or Algebra

A system consisting of a set and one or more n-ary operations on the set will be be called an algebraic system or simply an algebra.

We shall denote an algebraic system by (S, f₁,f₂, ...) where S is a nonempty set and f₁,f₂, are operations on S.

Definition 2: Algebraic structure

The operations and relations on the set S define a structure on the elements of S, an algebraic system is called an algebraic structure.

Example: Let I be the set of integers. Consider the algebraic system (1, +,˟) where + and ˟ are the operations of addition and multiplication on I.

A list of important properties

(A-1) For any a, b, c ϵ I

(a + b) + c = a + (b + c)        (Associativity)

(A-2) For any a, b ϵ I

a + b = b + a                 (Commutativity)

(A-3) There exists a distinguished element 0 Є I such that for any a Є I

a + 0 = 0 + a = a                (Identity element)

Here a Є I is the identity element with respect to addition.

(A-4) For each a Є I, there exists an element in I denoted by -a and called the negative of a such that

a + (-a) = 0           (Inverse element)

(M-1) For any a, b, c Є I

(a x b) x c = a x (b x c)        (Associativity)

(M-2) For any a, b Є I

a x b = b x a      (Commutativity)

(M-3) There exists a distinguished element 1 Є I such that for any a Є I

a x 1 = 1 x a = a         (Identity element)

(D) For any a, b, c Є I

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

The operation x distributes over +.

(C) For a, b, c Є I and a ≠ 0

a x b = a x c ⇒ b = c            (Cancellation property)

The algebraic system (I, +, x) should have been expressed as (I, +, ˟, 0, 1) in order to emphasize the fact that 0 and 1 are distinguished elements of I.

Definition 3: Homomorphism

If {X, o} and {Y, *} are two algebraic systems, where o and * are binary (n-ary) operations, then a mapping g: X→Y satisfying

8(x₁ o x₂) = g (x₁) * g (x₂) ∀ x₁, x₂ Є X is called a homomorphism.

Note: Let g be a homomorphism from (X, o) to (Y, *)

(i) If g X → Y is one-to-one, then g is called a monomorphism.

(ii) If g: X → Y is onto, then g is called a epimorphism.

(iii) If g: X → Y is one-to-one and onto, then g is called a isomorphism.

(iv) A homomorphism g: X → Y is called an endomorphism, if Y≤X

(v) An isomorphism g: X→ Y is called an automorphism, if Y=X

Definition 4:

Let (X, *) be an algebraic system and E be an equivalence relation on X. The relation E is called a congruence relation on (X, *) if E satisfies the substitution property with respect to the operation o.

Note: Substitution property

Let {X, *} be an algebraic system in which * is a binary operation on X. Let us assume that E is an equivalence relation on X.

The equivalence relation E is said to have the substitution property w.r.to the operation * iff for any x₁, x₂ Є X

(x₁ E x₁') ˄ (x₂ E x₂') = (x₁ * x₂) E (x₁' * x₂')

where x₁, x₂' Є X

Example 1. Show that intersection any two congruence relation on a set A is again an congruence relation on A.               [A.U N/D 2014]

Solution :

Let E₁ and E₂ be two congruence relations on (A, *)

⇒ (a₁ E₁ a₁') ^ (a₂ E₁ a₂') = (a₁ * a₂) E₁ (a1' * a₂')

(a₁ E₂ a₁') ^ (a₂ E₂ a₂') = (a₁ * a₂) E₂ (a₁' * a₂')

Let E = E₁ Ո E₂

To prove E is a congruence relation on A.

(a₁ E a₁') ^ (a₂ E a₂')

= [a₁ (E₁ ∩ E₂) a₁'] ^ [a₂ (E₁ ∩ E₂) a₂']

= (a₁ E₁ a₁') and (a₁ E₂ a₁) ^ (a₂ E₁ a₂') and (a₂ E₂ a₂')

= (a₁ E₁ a₁') ˄ (a₂ E₁ a₂') and (a₁ E₂ a₁) ^ (a₂ E₂ a₂').

= (a₁* a₂) E₁ (a₁' *a₂') and (a₁ * a₂) E₂ (a₁' * a₂')

= (a₁* a₂) (E₁ ∩ E₂) (a₁' * a₂')

= (a₁ * a₂) E (a₁' * a₂')

Hence, E is a congruence relation on A.

Example 2. Let f: S → T be a homomorphism from (S, *) to (T, ∆) and g: T → P is also a homomorphism from (T, ∆) to (P, V), then g o f: S → P is a homomorphism from (S,*) to (P, V).

Solution: As g o f (S₁ * S₂) = g(f(S₁ * S₂))

= g (f (S₁ ∆ f (S₂))     [Since ƒ is [homomorphism]

= (f(S₁ ∆ g (f (S₂))) [Since g is homomorphism]

= g o f (S₁) ∆ g o f (S₂)

= g o f: S → T is a homomorphism

Example 3. Let (A, *) and (B, ∆) be two algebra systems and be homomorphism from A→ B. Let (A₁, *) be subalgebra of (A, *). Then show that an the homomorphic image of (A₁, *) is a subalgebra of (B, ∆)

Solution: Let g be an homomorphism from A to B. Then for any two elements a₁, a₂, Є A.

g (a₁ * a₂) = g(a₁) ∆ g (a₂). Let A₁ be a subset of A. As g is homomorphism from A to B, for any two elements, a_j, a_j Є A₁ ≤ A.

g (a_i * a_j) = g(a_i) ∆ g(a_j) and g(A₁) ≤ g (A) ≤ B. Therefore the image of A₁ and g forms an algebraic system with operation ∆, which becomes a subalgebra of B.

Example 4. Give an example for homomorphism.      [A.U N/D 2014]

Solution: Let S = {a, b, c) and P = {1, 2, 3} be two sets with operation + and *

Consider the mapping g: S→ P defined by

g (a) = 3, g (b) = 1, g (c) = 2

g (a + b) = g(b) = 1 = 3*1 = g(a) * g(b)

g (b + c) = g(c) = 2 = 1*2 = g(b) * g(c)

g (c + a) = g(c) = 2 = 2*3 = g(c) * g(a)

g is homomorphism of {S, +} {P, *}

EXERCISE 4.1

1. Show that the composition of two congruence relations on a set is not necessarily a congruence relation.

2. Show that the intersection of any two congruence relations on a set is also a congruence relation.

3. Let S {a, b, c} and P = {1, 2, 3} be two sets with operations + and * defined on S and P respectively as given in Table

Show that (S, +) and (P, *) are isomorphic.

4. Given two algebraic system (W, +) and (Z₄, +₄) where W is the set of all non-negative integers and + is the usual addition operation defined on W. Then show that there is a homomorphism from W to Z₄.

5. Let (W, +) be an algebraic system of non-negative integers, where + is the usual addition. Define an equivalence relation R on W such that n₁ Rn₂ if and only if either n₁ - n₂ or n₂ - n₁ is divisible by 5. Show that R is an equivalence relation and that the homomorphism g defined from (W, +) to Z₅ by g(i) = [i] is the natural homomorphism associated with R.