On Smarandache Semigroups

In this work we study some type of Smarandache semigroups and Smarandache subgroups of a semigroup such as Smarandache cyclic semigroups, Smarandache pSylow subgroups and Smarandache normal subgroups. In addition we introduce the concept of Smarandache ideal of a semigroup and study its relation with Smarandache normal subgroup. Introduction A semigroup S called a Smarandache semigroup if there is a proper subset of S which is a subgroup of S (Raual, 1998), (by a subgroup A of S we mean a subset A of S which is a group under the same operation of S). It is known that if e is an idempotent of a semigroup S then Ge= {aS| a=ae and e=a1 a=a a1 for some a1S} equal to S or it is the maximal subgroup of S having e as its identity (Mario, 1973). Many Smarandache concepts introduced by Kandasamy,V. W. and many open research problems are given(Kandasamy, 2002). A Smarandache semigroup S called Smarandache cyclic semigroup if every subgroup of S is cyclic (Kandasamy, 2002). If S be a finite Smarandache semigroup, P a prime which divides the order of S, then a subgroup of S of order p or p t (t >1) called Smarandache p-Sylow subgroup. In this work we give complete answer of the following problems given in (Kandasamy, 2002). 1Find condition on n, n a non prime so that Zn, the semigroup under multiplication modulo n is a Smarandache cyclic semigroup. 2Let (Z2 n ,.) be the semigroup of order 2 n . For n>3 arbitrarily large find the number of Smarandache 2-Sylow subgroup of Z2 n . In addition we introduce the concepts of Smarandache ideal, Smarandache prime ideal and study some of their properties and we give the relation between Smarandache ideals and Smarandache normal subgroups. Journal of Kirkuk University – Scientific Studies , vol.6, No.1,2011 92 S1: Smarandache cyclic semigroups In this Section we discuss Smarandache cyclic semigroups, and find the number of cyclic subgroups of (Zp n ,.) for n>2. Lemma1.1. ( p n ,.) p prime, has no nontrivial idempotent. Proof: The proof is easy. Theorem 1.2. ( p n ,.) p an odd prime, n>2, is a Smarandache cyclic semigroup. Proof: Since 1 ) ( n n n p p p the number of elements in which have inverses form a group under multiplication, and then have a subset which is a group of order 1 n n p p . This subgroup is the largest subgroup with 1 as its identity. Since there exists an element which is a primitive root of (Kenneth, 2004), ) (mod 1 1 n p p p a n n and a generates , thus is cyclic. Hence all subgroups of p n are cyclic, and p n is a Smarandache cyclic semigroup. Lemma 1.3. Let (G,.) be a semigroup with identity 1and S={xG: x 2 =1}. Then (S,.) is a cyclic group if and only if S contains at most two elements. Proof: The proof is easy. Proposition 1.4. 1The semigroup ,.) ( 2 Z , k 2 is a Smarandache semigroup which is not a Smarandache cyclic semigroup. 2The semigroup ,.) ( 2 p k Z , k≥2, p an odd prime, is a Smarandache semigroup which is not a Smarandache cyclic semigroup. Proof: 1Since 1 2 ) 1 2 )( 1 2 ( , 1 ) 1 2 ( ) 1 2 ( 1 1 2 1 2 1 k k k k k , and 1 ) 1 2 ( 2 k , then )} 1 2 ( ), 1 2 ( ), 1 2 ( , 1 { 1 1 k k k S is a subgroup of ,.) ( 2 Z and by Lemma 1.3, S is not cyclic. Hence ,.) ( 2 Z is not a Smarandache cyclic semigroup. 2Similar to part 1. Theorem1.5. ,.) ( 2 n p Z , p odd prime is a Smarandache cyclic semigroup. Proof: First we show that n p Z2 has two maximal subgroups of order ) 2 ( n p . It is known that there exists a number belonging Journal of Kirkuk University – Scientific Studies , vol.6, No.1,2011 93 to ) 2 )(mod 2 ( n n p p , so ) 2 (mod 1 ) 2 ( n n p p , and a generates a group (G1) of order ) 2 ( n p with 1 as its identity. Since ) 2 1 ) 2 ( n n kp p for some 1 k , then n n n n kp p p p 2 ) 1 ( ) 2 ( . Therefore ) 2 )(mod 1 ( ) 2 ( n n n n p p p p . We claim that n p a generates a group of order ) 2 ( n p and n p 1 is its identity element. ) 2 )(mod ( ) ( 2 n n n p p p and ) 2 )(mod 1 ( ) 1 ( 2 n n n p p p , hence ) 2 )(mod ( ) ( 2 2 n n n p p a p a and ) 2 )(mod ( ) ( 3 3 n n n p p a p a . If a is even, then n n p ap , consequently ) 2 )(mod ( ) ( 3 3 n n n p p a p a . If a is odd, then ) 2 (mod n n n p p ap which implies that ) 2 )(mod ( ) ( 3 3 n n n p p a p a . Continuing in this manner we get ) 2 (mod 1 ) ( ) ( n n p n p p p a n , and ) 2 (mod ) ( 1 ) ( n n p n p p a p a n . This means that ( n p a )generates a subgroup of order ) 2 ( n p , and since n l n n l p a ) p 1 ( ) p a ( + = + + , for each ) ( 1 n p l then ) 1 ( n p is the identity element of the group generated by a+p n which is cyclic (the group G1+p n ) . Note that is a subgroup of n p Z2 . Since the maximal subgroups are cyclic, n p Z2 is a Smarandache cyclic semigroup. Proposition 1.6. ,.) ( m q p Z , where p,q are odd primes, is a non cyclic Smarandache