Definition 2: An algebraic system is a nonempty set s in which at least one or more operations C1…Q1(k≥1),are defined. We denoted by s; Qis, QkI Z;+] ●[Z;+,2 ]is not an algebraic system
• Definition 2: An algebraic system is a nonempty set S in which at least one or more operations Q1 ,…,Qk (k1), are defined. We denoted by [S;Q1 ,…,Qk ]. • [Z;+] • [Z;+,*] • [N;-] is not an algebraic system
Definition 3: Let [S; and T; are two algebraic system with a binary operation. A function (p from s to T is called a homomorphism from S;* to T; o if cp(a*b=p(ao(b)for Va, bES
Definition 3: Let [S;*] and [T;•] are two algebraic system with a binary operation. A function from S to T is called a homomorphism from [S;*] to [T;•] if (a*b)=(a)•(b) for a,bS
Theorem 6.3 Let p be a homomorphism from S; to T;. If p is onto, then the following results hold ()If x is Associative on S, then is also Associative on a (2)If x is commutative on S, then is also commutation on T 3)If there exist identity element e in S; l, then cp(e)is identity element of T (4)Let e be identity element of [s;. If there is the inverse element al of dES, then op(a-l)is the inverse element (p(a
Theorem 6.3 Let be a homomorphism from [S;*] to [T;•]. If is onto, then the following results hold. (1)If * is Associative on S, then • is also Associative on T. (2)If * is commutative on S, then • is also commutation on T (3)If there exist identity element e in [S;*],then (e) is identity element of [T;•] (4) Let e be identity element of [S;*]. If there is the inverse element a -1 of aS, then (a -1 ) is the inverse element (a)