12 Algebraic Structure and Coding Theory

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1 12.1 The Structure of Algebra Def 1 Let S be a set and be a binary operation on S( : S S S). 2. The operation is associative over S, if a (b c) = (a b) c. 1. The operation is commutative over S, if a b = b a. Def 2 Let S be a set and S S. Let be a binary operation on S and be an unary operation of S. Then 1. S is closed w.r.t., if a, b S, a b S. 1. S is closed w.r.t., if a S, a S Let us denote an algebra(algebraic system) (S, O, C) where (i) S is an underlying set, (ii) O is the set of operations, and (iii) C is the set of constants. 6/8/16 Kwang-Moo Choe 1

2 Def 3 Let A = (S, O, C) be an algebra. Then A = (S, O, C) is a subalgebra of A, if i) S S and ii) O O, o O : o is same as restricted to S. ( O s, t S s t S ) Ex. 7 (E, +, 0) is a subalgebra of (I, +, 0) but (O, +, 0) is not. Def 4 Let : S S S. An element e S is an identity(unit) element for the operation, if e s = s e = s, s S. An element 0 S is an zero element for the operation, if 0 s = s 0 = 0, s S. Exam. 9 0 Z is a identity element for the algebra (Z, +). 0 Z is a zero element and for the algebra (Z, ). 1 Z is a identity element for the algebra (Z, ). 6/8/16 Kwang-Moo Choe 2

3 Def 5 Let (S, ) be an algebra, e L S is a left identity element for the operation, if e L s = s, s S. 0 L S is a left zero element for the operation, if 0 L s = 0 L, s S. e R S is a right identity element for the operation, if s e R = s, s S. 0 R S is a right zero element for the operation, if s 0 R = s, s S. Thm1 Let : S S S with left identity e L and right identity e R. Then e L = e R and e = e L = e R is called as two-sided identity. proof e R = L e L e R = R e L. Thm 2 Let : S S S with left zero 0 R and right zero 0 R. Then 0 L =0 R and 0 = 0 L = 0 R is called as two-sided zero. proof 0 L = L 0 L 0 R = R 0 R. Cor 1 A two-sided identity(or zero) for a binary operation is unique. 6/8/16 Kwang-Moo Choe 3

4 Def 6 Let : S S S and e S is an identity for the operation, if x y = e, then x is the left inverse of y and y is the right inverse of y w.r.t. the operation. If both x y = y x = e, x is the inverse (or two-sided inverse) of y w.r.t., wrtitten y 1,. Thm. 3 If an element has both a left inverse and right inverse w.r.t. an associative operation : S S S. Then the left and right (or two-sided) inverse elements are equal. proof Let e be an identity element for the op.. Assume x S, y x = x z = e. y is a left inverse of x and z is an right inverse of x. y = y e = y (x z) = (y x) z = e z = z. y = z is a two-sided inverse of x. 6/8/16 Kwang-Moo Choe 4

5 12.2 Semigroups, Monoids, and Groups Def. 0 Algebraic System: (S, ) is an algebra(or algebraic system), if 1. S is an closed w.r.t a binary operation on : S S S. Ex. 0. ( 1, +) is an algrbra. i, j 1, i+j 1. Def 1 Semigroup: (S, ) is a semigroup, if 1. S is an algebraic system and : S S S, x,y S, x y S 2. is associative operation (x y) z = x (y z) = x y z. ( ((a 1 a 2 ) a 3 ) a n-1 ) a n left associative = a 1 (a 2 ( (a n-1 a n ) )) right associative = a 1 a 2 a n-1 a n = a 1 a 2 a n = a 1 a 2 a n. infix prefix postfix a a a a a a = a a a = a n. : S n S is an n-ary operator(n 0)(associative) a 1 a 2 a n = i 1,n a i (prefix) indexed(i) set(n 1,n ) notation 6/8/16 Kwang-Moo Choe 5

6 Ex. 1.1 ( 1, +) is a semigroup generated(++) by {1} 2 = 1 + 1, 3 = ,, n = , 2 = ++(1), 3 = ++(++(1)),, n = ++ n (1), Ex. 1.2 ( +, ) is a semigroup generated(concatenated( )) by a set of symbols in (vocabulary, alphabet; a set of symbols). We define a set of strings over V of length n( 1) as n = R n-1. x = (a 1 (a 2 (a n-1 a n ) )) V n, 1 i n, a i V. or n = R n-1. x = (( (a 1 a 2 ) a n-1 ) a n ) V n, 1 i n, a i V. The length of a string x is n, written x = n. We write x = a 1 a 2 a n (justtaxaposed) instead of (a 1 (a 2 (a n-1 a n ) )) = (( (a 1 a 2 ) a n-1 ) a n ) = (a 1 a 2 a n ) = (a 1, a 2,, a n ) = a 1 a 2 a n.. + = n 1 n = 2 n all strings of pos. length. n = n and + = n + =. Ex Let = {a, b,, z}. Then school 6, boy 3, schoolboy 9. 6/8/16 Kwang-Moo Choe 6

7 Def 2 Monoid: (S,, e) is a monoid, if 2. (S, ) is a semigroup and 3. e S is an identity element. Ex 2.1 ( 1, +,?) is not a monoid! But ( 0, +, 0) is a monoid. Ex = B { }. 0 (or ; empty string, = = 0): n = R n-1. x = a 1 a 2 a n-1 a n V n, 1 i n, a i V. We define the universe of strings over. * = n 0 n = { } +. universe of strings We define concatenation( ) of strings over *. : * * * x, y * : x y = xy (x y) z = x (y z) x * : x = x = x. is an identity w.r.t. (concatenation). * = =. ( *,, ) is a (free) monoid generated by the vocabulary 6/8/16 Kwang-Moo Choe 7

8 Def 3 Group: (S,, e) is a group, if ( 0, +, 0) is not a group 3. (S,, e) is a monoid and (Z, +, 0) is a group 4. x S, 1x 1 S(unique inverse) w.r.t x x 1 = x 1 x = e. We use to denote a group (S,, e, 1 ) instead of (S,, e) to specify the inverse binary operation 1 and unique inverse x 1. (, +, 0, ) is not a group but (Z, +, 0, ) is a group. x, y x y. But x, y Z, x y Z 1x 1 ( x) Z. Ex. 3 R = {r 0, r 60, r 120, r 180, r 240, r 300 } and : R R R r 1 r 2 = r where = [ ] 360. (R,, r 0, 1 ) is a group. 6. : Rotate clackwise 1 : Rotate counter clockwise. 6/8/16 Kwang-Moo Choe 8

9 G = (S,, e, 1 ) is a group, if 1. S is an closed w.r.t, 2. is an associative operation, 3. e S is an identity element w.r.t. the operation, and 4. x S, 1unique inverse element x 1 S w.r.t. 4.5 x, y S, x 1 y = x y 1 = e. Def. 3.5 Commutative (or Abelian) group 1. (S,, e, 1 ) is a group, and 2. is commutative. Ex. 4 (Z, +, 0, ) is a group. and (Z,, 1) is a monoid. Consider 3 congruence of modulus n operations 1 n, n, and n on Z n. n, 1 n, n : Z n Z n Z n where Z n = {0, 1,, n-1}, a n b = [a+b] n, a 1 n b = [a b] n, a n b = [a b] n. addi. cong. mod. of n, inverse of addi. of n, multi. mod. of n 6/8/16 Kwang-Moo Choe 9

10 Consider a subalgebra (Z n, { n, n 1, n }, {0, 1}). 1. Z n is closed under n and n ; i.e. a, b Z: a n b, a n b Z n. 2. n is commutative and associative. 0 is an identity of. 6. If a Z n, inverse of a w.r.t n, a 1 1 = n n a Z n. 3. (Z n, n, 0, 1 n ) is a group. 4. n is closed, commutative and associative. 1 is an identity of n. 4. a Z n, \ a 1 Z inverse of a w.r.t n a n a 1 = (Z n, n, 1) is a monoid but not a group. 5. n distributes over n. a n (b n c) = (a n b) n (a n c). If n 2, then (Z n, n, 0, n 1 ) is a finite (n 2) homomorphic image of (Aberian) group (Z, +, 0, ). (Z n, n, 1) a finite homomorphic image of a monoid (Z,, 1). 6/8/16 Kwang-Moo Choe 10

11 Finite Groups ({a}, {(a a = a)}, a) a group of order 1 ({a, b},, a) a group of order 2 ({a, b, c},, a) a group of order 3 ({a, b, c, d},, a) two groups of order 4 ({a, b, c, d, e},, a)??? groups of order 5 Finite Subgroup Thm. 1 Let (T,, e) be a group and T T. If T is finite, then (T,, e) is a subgroup of (T,, e), if T is closed under. proof Let a T. Then ( is associative and closed) a 2, a 3, a 4, a n, T. Since T is finite, a n = a m for some n, m 0, n m. a n = a n a m-n, m n 0. e = a m-n is an identity and in T. 6/8/16 Kwang-Moo Choe 11

12 Two cases for identity, e = a m-n, since T is finite. (2) If m n = 1, a 1 = e, a n+1 = a n a 1 = a n e = a n = a a = a a = a 2. a is idendty and inverse of a and in T. T = {a}. T is a singleton set. (1) If m n 1, a m-n = a a m-n-1 = e. a m-n-1 (m - n - 1 0) is the inverse of a and in T. T = {a 1, a 2,, a m-n-1, a m-n,, a m-1 }. T 2. 6/8/16 Kwang-Moo Choe 12

13 Generators for a group Let (T, ) be an algebraic system, and S T. Let S 1 = S {a b A a, b S}. S 1 is called the set generated directly by S. S 2 = S 1 {a b A a, b S 1 } S i+1 = S i {a b A a, b S i } S * = S S 1 S 2 c S *, a, b S *. a b = c. x S *, x is said to be generated by S. S * T is the subsystem generated by S. (S *, ) is called the subsystem generated by S. If S * is finite, (S *, ) is the subgroup.(theorem ) 6/8/16 Kwang-Moo Choe 13

14 If B * = A, B is called a generating set or a set of generators of the algebraic system (A, ). A group that has generating consisting of a single element is called as a cyclic group. Let (A, ) be a cyclic group with generating set {a}. A = {a, a 2, a 3, } a i a j = a j a j = a i+j. associative Any cyclic group is commutative group Let B be a generating set of an algebraic system(a, ). For a A, r 1, and a r = a. a 1 a 2 a r generating sequence for a A 1 i r, j, k i a i = a j a k. 6/8/16 Kwang-Moo Choe 14

15 Example) (, +, 0) is an commutative cyclic group generated by {1} with the identity 0. (V *,, ) is a (free) monoid generated by V with identity. Example) Consider (I, +), B = {1}, and consider 9 addition chain original addition chain for 9(++) shorter addition chain for 9 (4+5) shorter addition chain for 9(3+2, 5+3) the shortest addition chain for 9(3+3, 3+6) The shortest addition chain Method 1. If n = p q. If p 1 p 2 p i-1 p is the shortest addition chain(i) for p and q 1 q 2 q j-1 q is the shortest addition chain(j) for q. Then q 1 q 2 q j-1 q q p 2 q p 3 q p i-1 q p or p 1 p 2 p i-1 p p q 2 p q 3 p q i-1 p q. the shortest addition chain(j+i 1 = i+j 1) for n=p q 6/8/16 Kwang-Moo Choe 15

16 45 = 5 9 5: 1, 2, 3, 5 9: 1, 2, 4, 8, 9 45: 1, 2, 4, 8, 9, 18, 27, 45 or 1, 2, 3, 5, 10, 20, 40, 45 Method 2 If n is even, determine addition chain for n/2, n. If n is odd, determine addition chain for (n-1)/2, (n-1), n. 45, 44, 22, 11, 10, 5, 4, 2, 1 1, 2, 4, 5, 10, 11, 22, 44, 45 Method 2 Method Method 2 is semi optimal! 6/8/16 Kwang-Moo Choe 16

17 Cosets and Lagrange s Theorem Let (T, ) be an algebraic system. and H T. Then Left coset of H w.r.t. a T, denoted as, a H = {a x x H}. Right coset of H w.r.t. a T, denoted as, H a = {x a x H}. Thm. 4 Let a H and H a be two cosets of H. Then a H = H a or a H H a = Thm. 5 (Lagrange s Theorem) The order of any subgroup of a finite group devides the order of the subgroup. Thm. 6 Any group of prime order is cyclic and any element other than the identity is a generator. It also follows that it is abelian. 6/8/16 Kwang-Moo Choe 17

18 Isomorphism and Automorphism Two systems(algebra) (T, ) and (S, ) are isomorphic, if a bihective function f: T S, such that a, b T: f(a b) = f(a) f(b). An isomorphism from (T, ) to (T, ) is called automorphism. Permutation Group Consider S 3 = {1, 2, 3} be a set and a set of bijective mappings on S 3. Consider a triple (i, j, k) i, j, k S 3 and i j, j k, k i, (i, j, k) {(1, i), (2, j), (3, k)} = {f(1) = i, f(2) = j, f(3) =k} Then there are 6 (=3!) triples, P 3 = {(1, 2, 3) (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)} Consider composition of permutation : P 3 P 3 s.t. p q(i) = p(q(i)) Ex.) (2, 3, 1) (3, 1, 2) = (1, 2, 3), (2, 3, 1) (3, 2, 1) = (1, 3. 2) 6/8/16 Kwang-Moo Choe 18

19 Consider S n = {1, 2,, n} and P n = {p: S n S n }. Then (S n, ) is called as a permutation group of degree n. : P n P n P n. where p q(i) = p(q(i)); Note that S n = n, P n = n!, and = n! n! = n! 2. Thm. 7 Any finite group of order n is isomorphic to a permutation group of degree n. 6/8/16 Kwang-Moo Choe 19

20 12.3 Homomorphisms, Normal Subgroups, and Congruence Relations Def. 1 Let A = (T,, c) and A = (T,, c ) be algebra, and h: T T h(a) h(b) = h(a b), h(c) = c. Then Since we consider h that is onto, T T. A is a homomorphic image(or abstract interpretation) of A under h. A is calles as a concretization(or refinement) of A under h. Exa. 1 A group (Z n, n, 0, 1 n ) and a monoid (Z n, n, 1) are abstract interpretations of the group (Z, +, 0, ) and a monoid (Z,, 1) under h n (=[] n ). Exa. 2 A group ({ 홀, 짝 },, 짝, 1 n ) and a monoid ({ 홀, 짝 },, 홀 ) are abstract interpretation of (Z, +, 0, ) and (Z,, 1), respectively under h 2 (=[] 2 ). Exa. 2.1 The group ({ 홀, 짝 },, 짝, 1 n ) and a monoid ({ 홀, 짝 },, 홀 ) are isomorphic to (Z 2 = {0, 1}, +, 0, ) and (Z 2,, 1) under i(=[] 2 ). where i(0) = 짝 and i(1) = 홀 ; i 1 ( 짝 ) = 0 andi 1 ( 홀 ) = 1. 6/8/16 Kwang-Moo Choe 20

21 Def. 1 Let (T,, c ) be a homomorphic image of (T,, c) under h. Then we define a congruence relation ~ T T under h as a ~ b, if h(a) = h(b). Col. 1 The congruence relation ~ is equivalent. Thm 3 Let (T,, c ) be a homomorphic image of (T,, c) under h and the (equivalent) congruence relation under h w.r.t is ~. Then a, b T, a ~ b c ~ d h(a b) = h(c d). Exa. 3 Consider a group (Z, +, 0, ) and its homomorphic image (Z n, n, 0, n 1 ) under [] n. Then 3 ~ n+3 2n+5 ~ 5 [2n+8] n = [n+8] n. (Eqivalent) Congruence Class (Eqivalent) Congruence Partition a T: [a] ~ = [a] h = h(a) T. Par ~ (T) T. 6/8/16 Kwang-Moo Choe 21

22 12.4 Rings, Integral Domain, and Fields The operation is distributes over, if a (b c) = (a b) (a c). Def. 1 Let (A,, ) be an algebraic system with two operators and. Then (A,, ) is called as a ring, if 1. (A, ) is an abelian group, 1.5 (A,, e, 1 ) is a group, 2. (A, ) is a semigroup, and 3. The operation is distributive over. We may use ((A,, 0, 1, ) to denote a ring instead of (A,, ) Ex. (,, 0,, ) is not a ring, But (Z,, 0,, ) and (Z n, n, 0, n 1, n ) are rings. : additive operation of the ring a b: sum of a and b : multiplicative operation of the ring a b: multiplication of a and b 6/8/16 Kwang-Moo Choe 22

23 Let (A,, e, 1, ) is a ring with additive identity 0. a A, 0 a = (0 0) a = (0 a) (0 a) 0 a = 0 = a 0. 0: multiplicative zero as well as additive identity. Def. 2 (A,, 0, 1, ) is called an integral domain, if 1. (A,, 0, 1, ) is a ring, and 2. (A, ) is commutative semigroup, and If c 0 and c a = c b, then a = b where 0 is the additive identity and/or multiplicative zero. Ex. 2 Consider is set of integers Z. (Z,, 0,, ) is the integral domain Def. 3 (A,, 0, 1,, 1, 1 ) is called a field, if 1. (A,, 0, 1, ) is a integral domain. 2. (A {0},, 1, 1 ) is an abelian group. 6/8/16 Kwang-Moo Choe 23

24 (A {0},, 1, 1 ) is a group 1: multiplicative identity a 1 : multiplicative inverse of a A {0} a a 1 = 1. 1 :A A {0} A a A, b A {0}: a 1 b = a ( 1 b) multiplication of a and multiplicative inverse of b Example) Let Q be the set of rational numbers. Then (Q,, 0,,, 1, /) is a field. Let is the set of real numbers. Then (,, 0,,, 1, /) is a field. Let C is the set of complex numbers. Then (C,, 0,,, 1, /) is a field. Substruction is not really an independent operation but it is the addition of the additive inverse. Division is the multiplication of the multiplcative inverse. 6/8/16 Kwang-Moo Choe 24

25 12.5 Quotient and Product Algebras Consider (Z n, n, n ) a n b = the remainder of a+b divided by n. a n b = the remainder of ab divided by n. (Z n, n ) is an abelian group, and is commutative. (Z n {0}, n ) is an abelian group iff n is prime. proof If n is not prime, n = ab for some a, b Z n {0}, But a, b Z p {0}, a n b = 0 not closed If n=p is prime, a, b Z p {0}, a b 0 a p b Z p {0}. p is closed under Z p {0}. p is associative and commutative 1 is the identity for p. If a, b c Z p {0}, then a p b a p c. 6/8/16 Kwang-Moo Choe 25

26 proof assume a p b = a p c ab = kp + r, ac = lp + r a(b c) = (k l)p (assume b c, k l) Since a, b c n, and p is prime a(b c) (k-l)p. a p b a p c a Z n {0}, b Z n {0}. a b = /a(= a) = b. (Z n -{0},, ) is a field. field of integers modulus n 12.6 Coding Theory 12.7 Polynomial Rings and Polynomial Codes 6/8/16 Kwang-Moo Choe 26

12 Algebraic Structure and Coding Theory

12 Algebraic Structure and Coding Theory 12.1 The Structure of Algebra Def 1 Let S be a set and be a binary operation on S( : S S S). 2. The operation is associative over S, if a (b c) = (a b) c. 1. The operation is commutative over S, if a b