The notions of the SMARANDACHE GROUP and the SMARANDACHE BOOLEAN RING

unpublished
The notions of the Snmarandache group and the Smarandache Boolean ring are introduced here with the help of group action and ring action i.e. module respectively. The centre of the Smarandache groupoid is determined. These are very important for the study of Algebraic structures. 1. The centre of the Smarandache groupoid: Definition 1. 1 An element a of the smarandache groupoid (Zp, Ll) is said to be conjugate to b if there exists r in Zp such that a = r Ll b Ll r. Definition 1.2 An element a
more » ... 1.2 An element a of the smarandache groupoid (Zp, Ll) is called a self conjugate element of Zp if a = r Ll a Ll r for all r E Zp. Definition 1.3 The set Zp* of all self conjugate elements of (Zp, Ll) is called the centre of Zp i e Zp* = { a E Zp : a = r Ll all r 'r:;j r E Zp}. Definition 1.4 Let (Z9' +9) be a commutative group, then Z9={ 0, 1,2,3,4,5,6, 7, 8 }. If the elements of Z are written as 3-adic numbers, then 9 Z9 = { (00)3' (01)3' (02)3' (10)3' (11)3' (12)3' (20)3' (21)3' (22)3 } and (Z9' Ll) is a smarandache groupoid of order 9. Conjugacy relations among the elements of Z are determined as follows: o 16
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