Two Divisibility Tests for Smarandache semigroups are given. Further, the notion of divisibility of elements in a semigroup is applied to characterize the Smarandache semigroups. Examples are provided for justification

Rao, Chandra Sekhar

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 1DIVISIBILITY TESTS FOR SMARANDACHE SEMIGROUPS.  AKS Chandra Sekhar Rao. Chandhrasekhar_6@yahoo.co.in Department of Mathematics D.N.R.College (Autonomous) Bhimavaram – 534 202 A.P.,India  Abstract  Two Divisibility Tests for Smarandache semigroups are given .  Further, the notion of divisibility of  elements in a semigroup is applied to characterize the Smarandache semigroups.  Examples are provided for justification.   Key words : Semigroup , Divisibility, Right Divisor , Left Divisor , Group , Smarandache semigroup. 1.Introduction  Padilla Raul introduced the notion of Smarandache semigroups in the year 1998 in the paper entitled Smarandache algebraic structures[2].  Since groups are the 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.  The Smarandache notions in groups and the concept of Smarandache semigroups have been studied in [9].  In[7] , the notion of divisibility of elements in  a semigroup is introduced and the properties of the elements which are both left and right divisiors of every element of a semigroup are studied. Further , the properties of the elements which are divisible both on the right and on the left by all elements of the semigroup are studied as well.  Such elements were first considered in a paper by Clifford and Miller, where they were termed zeroid elements.  2 The concept of divisibility of elements in a semigroup is very much useful to study the Smarandache semigroups.  In this paper we give two tests called Divisibility tests for Smarandache semigroups , and a characterization of Smarandache semigroups by applying the notion of divisibility of elements in a semigroup.  Examples are provided for justification.  In section 2 we give the basic definitions and properties of divisibility of elements in a semigroup.  The definition and example of Smarandache semigroups are given as well.  In section 3 we present our theorems and in section 4 we provide examples for justification.  For , more basic definitions and concepts please refer [7]and [9].  In this paper , we denote the operation multiplication (product)of elements in a semigroup by jux-ta-position. 2.Preliminaries Definition 2.1([7]). A semigroup is a nonempty set S, in which for every ordered pair of elements x , y  in S there is defined a new element called their product xy∈S, where for all x , y , z ∈S , we have (xy)z = x(yz). Definition 2.2([7]).  An element b of the semigroup S is called a right divisor of the element a of  the semigroup if there exists in S an element x such that xb = a.  b is called a left divisor of a if there exists in S an element y such that by = a. If b is a right divisor of a , we say that a is divisible on the right by b.  If b is a left divisor of a , we say that a is divisible on the left by b.  The following observations in the divisibility of elements in a semigroup are well known in [7].   32.2.1 An element a of a semigroup S will be a right (left) divisor of every element of S if and only if in the column (row) of the multiplication table corresponding to the element a , all elements of S occur.  2.2.2 The element a is divisible on the left(right)by every element of S if and only if a occurs in every row(column) of the multiplication  table.  Definition 2.3.([7]).  A nonempty subset H of the semigroup S is called a subsemigroup of S if HH⊂S. Definition 2.4.([6]).  A nonempty set G, together with an associative binary operation *  on G such that equations a * x=b and y * a= b have solutions in G for all a, b ∈ G , is a group.  Several examples of Semigroups, subsemigroups and groups can be found in the literature.  Definition 2.5([9]).  The Smarandache semigroup defined to be a semigroup S such that a proper subset B of S is a group with respect to the same operation on S. Example 2.6. Let us consider the semigroup S={0,1,2,3,4,5} under the operation multiplication modulo 6.  The multiplication table is as follows :  0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 2 3 4 5 2 0 2 4 0 2 4 3 0 3 0 3 0 3 4 0 4 2 0 4 2 5 0 5 4 3 2 1 Table 1.  4 S is a Smarandache semigroup as the proper subsets { 2 , 4 } , { 1 , 5 } of S are groups. 3. PROOFS OF THE THEOREMS  In this section we present Divisibility test 1 , Divisibility test 2 for Smarandache semigroups. Further, a Characterization of Smarandache semigroups, using the notion of  divisibility of elements in a semigroup, is given as well . Theorem 3.1 (Divisibility test – 1)   A semigroup S ( not a group ) containing the elements that are both left and right divisors of every element of S is always a Smarandache semigroup. Proof :  Let G be the set of all elements that are both left and right divisors of every element of S.  Cleary G ⊂S.  ( if S = G , then G is a group as every element of  S is a left and right divisors of every element of S.).  To show that S is a Smarandache semigroup, it is sufficient to show that G is a group under the operation on S. Suppose that b1, b2 ∈   G.  For every a ∈ S there exists x2 in S  such that       x2b2 = a.  Corresponding to x2 ∈  S there exists in S an element x1 such that x1b1 = x2 .  Therefore,  x1 b1b2 = x2 b2 = a.    Accordingly , b1b2 is a right divisor of a .   Similarly , we can show that b1b2 is a left divisor of a.  .  Therefore, b1b2 ∈  G.  The multiplication in G is associative as G ⊂  S.  Suppose that b1 , b2 ∈  S .   There exist in S elements x and y such that xb2 = b1 ,    b2 y= b1 .  If we show that x , y ∈  G the proof that G is a group will be completed.   Let  α  be an arbitrary element of S.  Since, for some t ∈ S, we have b1t =α  ; xb2t = b1t = α  .  i.e., x is a left divisor of α   There exist also in S elements e , c1 , c2 such that b2e = b2, c1b1 = e , b2c2 = e.  Since for arbitrary z in S there exists an element z1 ∈  S such that z = z1b2 , we have                 5ze = z1b2e = z1b2 = z   i.e., e is such that ze = z  for arbitrary z ∈S.   Making use of this property of e ,       we obtain, α = α e= α b2c2=α b2ec2 =α b2c1b1c2=α b2c1 x b2c2 = α b2c1xe = α b2c1x. i.e., x  turns to be also a right divisor of α  .  Therefore, x ∈ G.  Analogously, we can show that y ∈  G.  Hence, S is a Smarandache semigroup. Theorem 3.2 (Divisibility test – 2) A semigroup S ( not a group) containing the elements that are divisible both on the right and on the left by every element of S is always a Smarandache semigroup. Proof : Let G be the set of all elements of S that are divisible both on the right and on the left by every element of S.  Clearly , G ⊂  S. Let c1 , c2∈G, for every a ∈  S , there is some x in S such that c1 = ax.  Therefore,    c1c2 = a(xc2), i.e., c1c2 is divisible on the left by an arbitrary element a.  Analogously, we can show that c1c2 is divisible on the right by a.  Therefore, c1c2∈  G.  The associativity of the operation in G follows from the fact that G ⊂  S.  Since c1 and c2 belong to G, there exist elements u ,v ,  w , in S such that  uc22 = c2 , c22v = c2 ,     c1 = c2w ,       We have then uc2 = uc2c2v = c2v .  This yields, c2(c2v2c1) = (c22v)(vc1) = c2vc1= uc2c1 = uc2c2w = c2w = c1. We have shown that the element y = c2v2c1 is a solution of the equation        c2y = c1. Let us show that  y ∈  G.  In fact , for arbitrary z in S there exist u1 and v1in S such that c1 = u1z, c2 = zv1.  Therefore, y = c2v2c1 = zv1v2u1z.   i.e.,  y is divisible by z both on the  right and on the left.  Analogously, we may find a solution x of the equation xc2 = c1, such that x ∈ G.  Therefore, G is a group .  Hence, the semigroup S is  a Smarandache semigroup.  6Theorem 3.3 .  Let S be a semigroup.  S is a Smarandache semigroup if and only if there exists a proper subsemigroup T of S such that T possesses an element which is both a right and left divisor of every element of T and is at the same time itself divisible both on the left and on the right by every element of T. Proof : Let S be a semigroup.   If S is a Smarandache semigroup then there exists a proper subset T of S such that T is a group with respect to the same operation on S.  T possesses an element which is both a right and a left divisor of every element of T and is at the same time itself divisible both on the left and on the right by every element of T as T is a group with respect to the same operation on S.  On the other hand suppose that there exists a subsemigroup T of S such that T possesses an element which is both a right and a left divisor of every element of T and is at the same time itself divisible both on the left and on the right by every element of T.  Now we show that S is a Smarandache semigroup .  For, it is sufficient  to prove that the proper subsemigroup T is a group.  Let a and b be arbitrary elements of T and let d be an element having the properties postulated in the hypothesis of the theorem.  There exist elements x1 , y1 , x2 , y2 in T such that a = x1d ,  a = dy1,  d = x2b,  d = by2 It follows that x = x1x2 and y = y1y2 are solutions of the equations a = xb ,       a = by in T.  Therefore, T is a group and hence S is a Smarandache semigroup.   4. Examples   In this section we provide examples to justify our Divisibility test 1 , Divisibility test 2 for Smarandache semigroups , and  the characterization of Smarandache semigroups as well .  Further , we give examples to show that the conditions stated in theorem (3.1) and theorem (3.2 ) are sufficient conditions but not necessary conditions .  7 Example 4.1  Let z8 = {0,1,2,3,4,5,6,7} be a semigroup under multiplication mod 8 .  The composition table is as follows.           Table 2 In view of (2.2.1) the elements 1 and 7 in z8 are both left and right divisors of every element of the semigroup z8.  So, in view  of the theorem(3.1), the proper subset        G = {1,7} is a group under the multiplication mod 8.  Hence, the semigroup (z8 , x8)is a Smarandache semigroup.         X8 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 0 2 4 6 0 2 4 6 3 0 3 6 1 4 7 2 5 4 0 4 0 4 0 4 0 4 5 0 5 2 7 4 1 6 3 6 0 6 4 2 0 6 4 2 7 0 7 6 5 4 3 2 1  8Example 4.2  Let z7={0,1,2,3,4,5,6,}be a semigroup under multiplication mod 7.  The composition table is as follows. X7 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1  Table 3 In view of (2.2.1) the elements 1,2,3,4,5 and 6 in z7 are both left and right divisors of every element of the semigroup z7.  So, in view of the theorem (3.1), the proper subset G={1,2,3,4,5,6} is a group under the multiplication mod 7.  Hence, the semigroup   (z7 , x7)is a Smarandache semigroup.            9Example 4.3  Let z4={0,1,2,3} be a semigroup under the multiplication mod 4.  The composition table is as follows.        Table 4 In view of (2.2.1) the elements 1 and 3 in z4 are both left and right divisors of every element of the semigroup z4.  So, in view of the theorm (3.1) , the proper subset G={1,3} is a group under the multiplication mod 4.  Hence, the semigroup (z4,x4) is a Smarandache semigroup. Example 4.4  Let S={e,a,b,c} be a semigroup under the operation defined by the following composition table.  e a b c e e a b c a a e b c b b b c b c c c b c  Table 5 In view of (2.2.2) the elements b and c are divisible both on the right and on the left by every element of the semigroup S.  In view of the theorm (3.2), the proper subset X4 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1  10G={b,c} of S is a group under the operation on S.  Hence, S is a Smarandache semigroup. Example 4.5 Let S = { 0 ,1,2,3,4,5,6,7,8} be a semigroup under multiplication mod 9.  Consider the proper subset T = {1,2,4,5,7,8} of S.  The composition table on T of the multiplication mod 9 is as follows: X9 1 2 4 5 7 8 1 1 2 4 5 7 8 2 2 4 8 1 5 7 4 4 8 7 2 1 5 5 5 1 2 7 8 4 7 7 5 1 8 4 2 8 8 7 5 4 2 1  Table 6. From table 6 it is evident that T is a subsemigroup possessing the properties  postulated in the hypothesis of the theorem (3.3).  Therefore, T is a group and hence S is a Smarandache semigroup. Finally, we show by providing examples that the conditions stated in theorem (3.1) and theorem (3.2) are sufficient but not necessary .       11Example 4.6 Let S={1,2,3,4,5,6} be a semigroup under the operation defined by      xy = the great common divisor of x,y for all x, y ∈ S.  The composition table is as follows.         Table 7. This semigroup is not  satisfying the conditions stated in theorem (3.1) but this semigroup S is Smarandache semigroup . (see [1]).  Example 4.7  Let us consider the example 2.6 .  From the table 1 it is evident that the semigroup S in the example 2.6 is not satisfying the conditions stated in theorem (3.2) but this semigroup S is Smarandahce semigroup.    References  [1]  AKS Chandra Sekhar Rao , On Smarandache Semigroups , Scientia Manga, P.R. CHINA .(To appear ).   [2]Raul Padilla, Smarandache Algebraic Structures, Bulletin  of Pure and Applied Sciences , Delhi Vol 17 E(1998),No. 1 , 119-121. [3]T.Srinivsa, AKS Chandra Sekhar Rao, On Smarandache Rings, Scientia Magna, Vol 5 2009 , 117-124  1 2 3 4 5 6 1 1 1 1 1 1 1 2 1 2 1 2 1 2 3 1 1 3 1 1 3 4 1 2 1 4 1 2 5 1 1 1 1 5 1 6 1 2 3 2 1 6  12[4]Florentin Smarandache, Special Algebraic Structures in collected papers, vol III Abaddaba , oradea, (2000), 78 -81. [5] Nathan Jcobson, Basic Algebra, Hindustan Publishing Company , India, Vol I [6]John B. Fraleigh , A first course in Abstract Algebra , 3ed Addison – Wesley Publishing Company , Inc , USA [7] E.S.Ljapin, Semigroups , Translations of Mathematical Monographs, Vol.3 , American Mathematical society, providence, Rhode Island 1974. [8]W.B.Vasantha Kandasamy, Smarandache Semirings and Smarandache semifields , Smarandache Notions Journal, American Research press Vol 13, (2002), 88-91 [9]W.B.Vasantha Kandasamy , Smarandache Semigroups , American Research press , Rahoboth , NM , USA 2002. [10]W.B.Vasantha Kandasamy, Smarandache Rings, American Research press, Rehoboth , NM , USA 2002. [11]W.B Vasantha Kandasamy Groupoids and Smarandache groupoids, American Research press, Rehoboth , NM , USA  2002. [12]V.V.Vanger, The Theory of Generalized Heaps and Generalized groups, Mat.Sb.32(74)(1953), (/Russian), MR,15,501,545-632 

DIVISIBILITY TESTS FOR SMARANDACHE SEMIGROUPS

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