In this paper we introduce the concept of half-groups. This is a totally new
concept and demands considerable attention

Muktibodh, Arun S.

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International J.Math. Combin. Vol.2 (2008), 46-50Smarandache Half-GroupsArun S.Muktibodh(Mohota Science College, Umred Rd., Nagpur-440009, India.)E-mail: amukti2000@yahoo.comAbstract: In this paper we introduce the concept of half-groups. This is a totally newconcept and demands considerable attention. R.H.Bruck [1] has defined a half groupoid.We have imposed a group structure on a half groupoid wherein we have an identity elementand each element has a unique inverse. Further, we have defined a new structure calledSmarandache half-group. We have derived some important properties of Smarandache half-groups. Some suitable examples are also given.Key Words: half-group, subhalf-group, Smarandache half-group, Smarandache subhalf-Group, Smarandache hyper subhalf-group.AMS(2000): 20Kxx, 20L05.§1. IntroductionDefinition 1.1 Let (S, ∗) be a half groupoid (a partially closed set with respect to ∗ ) such that(1) There exists an element e ∈ S such that a ∗ e = e ∗ a = a, ∀a ∈ S. e is called identityelement of S;(2) For every a ∈ S there exists b ∈ S such that a ∗ b = b ∗ a = e (identity) b is called theinverse of a.Then (S,*) is called a half-group.Remark It is easy to verify that(a) identity element in S is unique;(b) each element in S has a unique inverse;(c) associativity does not hold in S as there is at least one product that is not defined in S.Note In all composition tables in the following examples the blank entries show that thecorresponding products are not defined.Example 1.1 Let S = {1,−i, i}. Then S is a half-group w.r.t. multiplication. We write thismultiplication table in the following.1Received March 2, 2008. Accepted March 28, 2008.Smarandache Half-Groups 47∗ 1 -i i1 1 -i i-i -i 1i i 1Example 1.2 Let S = {e, a, b, c}. Then (S, ∗) is a half subgroup defined by∗ e a b ce e a b ca a b c eb b c e ac c e aHere the product c ∗ c is not defined.Definition 1.2 it Let (S, ∗) be a half-group and H a subset of S. If H itself is a half-groupw.r.t. ∗, then H is called a subhalf-group of S.Example 1.3 Let S = {e, a, b, c, d} be a half-group defined by the following table.∗ e a b c de e a b c da a c e b ab b e c a dc c d a e bd d b c eThen , H = {e, a, b} is a subhalf-group of S.Definition 1.3 A half-group (S, ∗) is called a Smarandache half-group if S contains a propersubset G such that G is a nontrivial group w.r.t. ∗.Definition 1.4 If S is Smarandache half-group such that every group contained properly in Sis commutative, then S is called Smarandache commutative half-group.Definition 1.5 If S is a Smarandache half-group such that every group contained properly inS is cyclic, then S is called a Smarandache cyclic half-group.Example 1.4 Let S be a half-group defined by the following table.48 Arun S. Muktibodh∗ e a b ce e a b ca a e c bb b c e ac c e eThen G = {e, a} is a nontrivial group contained in S. So, S is a Smarandache half-group. Also,{e, a, b} is a Smarandache half-group. S is also a Smarandache commutative half-group. AlsoS is a Smarandache cyclic half-group.Example 1.5 S = {1,−i, i} is not a Smarandache half-group.Example 1.6 Let L be the Half-Group given by the following table.∗ e f g h i j k le e f g h i j k lf f e j g k h l ig g j e k h l i fh h g k e l i f ji i k h l e f j gj j h l i f e g kk k l i f j g el l i f j g k eThen L is a half-group which contains a group G = {e, g}. So, L is a Smarandache Half-Group.There are many Smarandache half-groups in this structure. Results following are obtainedimmediately by definition(1) The smallest half-group is of order 3.This follows from the very definition of half-groups.(2) The smallest Smarandache half-group is of order 3.As a nontrivial group has order at least 2, the half-group which will contain this groupproperly will have order at least 3.§2. Substructures of Smarandache Half-GroupsIn this section we introduce Smarandache substructure.Definition 2.1 Let S be a half-group w.r.t. ∗. A nonempty subset T of S is said to beSmarandache subhalf-group of S if T contains a proper subset G such that G is a nontrivialgroup under the operation of S.Smarandache Half-Groups 49Theorem 2.1 If S is a half-group and T is a Smarandache subhalf-group of S then S is aSmarandache half-group.Proof As T is a Smarandache subhalf-group of S , S contains T properly. Also, T properlycontains a non trivial group. As a result S is a hlf-group which properly contains a nontrivialgroup. Therefore S is a Smarandache half-group. We also note facts following on Smarandache half-groups.(1)If R is a Smarandache half-group then every subhalf-group of R need not be a Smaran-dache subhalf-group.We give an example to justify our claim.Example 2.1 Consider a half-group S defined by the following table.∗ e f g h i je e f g h i jf f h e g j ig g e h f i ih h g f e e ji i j i j ej j i f i eThen S ⊃ H = {e, f, g, h} and H is a group. Therefore S is a Smarandache half-group.Consider a half-group R = {e, f, g}. Then R is not a Smarandache subhalf-group of S as theredoes not exist a non trivial group contained in R.We give a typical example of a half-group following whose subhalf-groups are Smarandachesubhalf-group.Example 2.2 Consider the following table.∗ e f g h i j k le e f g h i j k lf f e j g k h l ig g j e k h l i fh h g k e l i f ji i k h l e f j gj j h l i f e g kk k l i f j g el l i f j g k eOne can easily verify that every subhalf-group is a Smarandache subhalf-group.50 Arun S. MuktibodhDefinition 2.2 If S is a Smarandache half-group such that a subhalf-group A of S contains thelargest group in S then A is called a Smarandache hyper subhalf-group.In the example above, the largest non-trivial group in S is of order 2 and every Smarandachesubhalf-Group of S contains the largest group in S. Thus, every Smarandache subhalf-Groupin S is a Smarandache hyper subhalf-Group.References[1] R.H.Bruck , A survey of Binary Systems, Springer Verlag,(1958).[2] W.B.Kandasamy, Smarandache nonassociative rings, Smarandache Notions Journal, Vol.14,p.281, (2004).[3] R. Padilla, Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences,Delhi, Vol.17 E. No.1, 119-121, (1998).

Smarandache Half-Groups

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