On The Smarandache Semigroups

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Journal of Kerbala University , Vol. 8 No.3 Scientific . 2010

On The Smarandache Semigroups ‫حٌل أشباه شمس سمازاندش‬ sajda Kadhum Mohammed lecturer Dep.of Math. , College Of Education for Girls, Al-Kufa University

Abstract: We discusse in this paper a Smarandache semigroups , a Smarandache normal subgroups and a Smarandache lagrange semigroup.We prove some results about it and prove that the Smarandache semigroup Z p n with multiplication modulo pn where p is a prime has the subgroup of order pn-pn-1and we prove that if p is an odd prime then

Z p n is a Smarandache weakly

Lagrange semigruop and if p is an even prime then Z p n is a Smarandache Lagrange semigruop ‫شمس سمازاندش الناظميةً أشبا ه شمس الكسانج سمازاندش حيث بسىناا‬, ‫ ناقشنا في ىرا البحث أشبا ه شمس سمازاندش‬:‫الخالصة‬ ٌ‫ عاد أًلاي جحا‬p ‫ حياث‬pn ‫ ما عملياة الباسع معيااز‬Z p n ‫كماا أببحناا أش شابو الصماس‬. ‫مجمٌعة من النحائج المحعلقة بيا‬ ‫ عاد أًلاي فاس فااش شابو الصماس أعا ه جكاٌش شابو‬p ‫ كما أببحنا اناو ذذا كاناث‬pn-pn-1 ‫مجمٌعة جصئية جكٌش شمس ذات زجبة‬ ‫ عااد أًلااي شًجااي فاااش شاابو الصمااس أع ا ه جكااٌش شاابو شمااس الكااسانج‬p ‫شمااس الكااسانج ساامازاندش ببااعا أمااا ذذا كانااث‬ .‫سمازاندش‬

1.Introduction : Padilla Raul introduced the notion of Smarandache semigroups[1], in the year 1998 in the paper entitled Smarandache Algebraic Structures .since groups are prefect structures under a single closed associative binary operation ,it has become infeasible to define Smarandache groups . Smarandache semigroups are the analog in the Smarandache ideologies of the groups where Smarandache semigroup is defined to be the semigroup A such that a proper subset of A is a group (with respect to the same binary operation). In this paper we prove some results in a Smarandache normal subgroups[1],[2], a Smarandache lagrange semigroups[1],[2],a Smarandache direct product semigroups[2],a Smarandache strong internal direct product semigeoups[2] , the S-semigroup homomorphism[1],[2] and prove research problems in the references about the semigroup Z p n we solve then using Mathlab programming to check our results in this open problems .

2.Definitions and Notations: Definition 2.1:The Smarandache semigroup (S-semigroup) is defined to be a

semigroup A such that a proper subset of A is a group (with respect to the same binary operation),[2]. Example 2.1: Let z12={0,1, … ,11} be the semigroup under multiplication module 12.Clearly the set A={1,11}  z12 is a group under multiplication modulo 12,so z12 is the Smarandache semigroup,[2] . Definition 2.2: Let S be a S-semigroup.Let A be a proper subset of S which is a group under the operation of S.We say S is a Smarandache normal subgroup of the S-semigroup if xA  A and Ax  A or xA={0} and Ax={0}  x  S and if 0 is an element in S then we have xA= {0} and Ax={0},[2].

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