1. Unit – IV Algebraic Structures
Algebraic systems
Examples and General Properties
Semi groups and monoids
Groups
Sub groups
Homomorphism
Isomorphism

2. 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 *.

3. 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.

4. 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.

5. 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)

6. * a b c d 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

7. 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.

8. Let * be a binary operation on nonempty set S.
Inverse (x)
Let * be a binary operation on nonempty set S.
xl * a = a * xr = e.
a is invertible.
Left Inverse (x1)
xl * a = e.
a is left-invertible.
Right Inverse (xr)
a * xr = e.
a is right-invertible.

9. 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.

10. 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.

11. 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.

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

13. 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.

14. 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.

15. Field If a ring <R, +, .> Examples
is commutative
has the unity
every 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.

16. 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. + 1 . 1

17. 2) Let S = {a, b, c, d} and the operations + and
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 b c d

18. 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.

19. 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 +.

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

21. 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

22. Consider the binary operation
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

23. In each of the following, indicate whether the given set forms a semigroup or a monoid under the given operation.
The set of all positive integers, with a * b = maximum of a and b.
The set of all even integers on which the operation * is defined by a * b = ab / 2.
The set A = {1, 2, 3, 6, 9, 18} on which the operation * is defined by a * b = LCM of a and b.
The set Q of all rational nos. on which the operation * is defined by a * b = a – b + ab.
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).

24. Let a nonempty set G be closed under *.
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).

25. 1. <Z, +> Abelian / Commutative Group Infinite Group Examples
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).

26. 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).

27. 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.

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

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

30. 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

31. 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+}

32. 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