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)

* 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

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.

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.

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.

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.

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, +, .> 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.

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

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

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 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.

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

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

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).

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).

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).

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.

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.

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.

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+}

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