By M. Sion

ISBN-10: 3540065423

ISBN-13: 9783540065425

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Bruck, A Survey of Binary Systems, Springer Verlag, (1958). 2. Vasantha Kandasamy, On Ordered Groupoids and its Groupoids Rings, Jour. of Maths and Comp. , Vol 9, 145-147, (1996). 3. Vasantha Kandasamy, New classes of finite groupoids using Zn, Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 44 CHAPTER FOUR SMARANDACHE GROUPOIDS In this chapter, we show the notion of SG, define and study several of its new properties. Here we work on new Smarandache substructures, identities on SGs and study them.

A non-empty subset H of G is said to be a Smarandache subgroupoid if H contains a proper subset K ⊂ H such that K is a semigroup under the operation ∗. 1: Every subgroupoid of a Smarandache groupoid S need not in general be a Smarandache subgroupoid of S. Proof: Let S = Z6 = {0, 1, 2, 3, 4, 5} set of modulo integer 6. The groupoid (S,∗) is given by the following table: ∗ 0 1 2 3 4 5 0 0 4 2 0 4 2 1 5 3 1 5 3 1 2 4 2 0 4 2 0 48 3 3 1 5 3 1 5 4 2 0 4 2 0 4 5 1 5 3 1 5 3 The only subgroupoids of S are A1 = {0, 3}, A2 = {0, 2, 4} and A3 = {1, 3, 5}.

To prove the converse we give the following example. 6, clearly (G, ∗) is a non commutative groupoid, but (G, ∗) is a Smarandache commutative groupoid. Hence the claim. 47 PROBLEM 1: Construct a groupoid of order 17, which is not commutative but Smarandache commutative. PROBLEM 2: How many SGs of order 5 exist? PROBLEM 3: Give an example of a Smarandache commutative groupoid of order 24 in which every semigroup is commutative. PROBLEM 4: Find a Smarandache commutative groupoid of order 12. PROBLEM 5: Find all semigroups of the groupoid Z8 (2, 7).

### A Theory of Semigroup Valued Measures by M. Sion

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