Zeroids and idempoids in AG-groupoids
Qaiser Mushtaq · 2004
Clifford and Miller (Amer. J. Math. 70, 1948) and Dawson (Acta Sci. Math. 27, 1966) have studied semigroups having left or right zeroids in a semigroup. In this paper, we have investigated AG-groupoids, and AG-groupoids with weak associative law, having zeroids or idempoids. Some interesting characteristics of these structures have been explored. An Abel-Grassman’s groupoid [8], abbreviated as AG-groupoid, is a groupoid G whose elements satisfy the left invertive law: (ab)c = (cb)a. It is also called a left almost semigroup [4, 5, 6, 7]. In [3], the same structure is called a left invertive groupoid. In this note we call it an AG-groupoid. It is a useful non-associative algebraic structure, midway between a groupoid and a commutative semigroup, with wide applications in the theory of flocks. AG-groupoid is medial [5], that is, (ab)(cd) = (ac)(bd) for all a, b, c, d in G. It has been shown in [5] that if an AG-groupoid contains a left identity then it is unique. It has been proved also that an AG-groupoid with right identity is a commutative monoid, that is, a semigroup with identity element. An element a◦ of an AG-groupoid G is called a left zero if a◦a = a◦ for all a ∈ G. It has been shown in [5] that if ab = cd then ba = dc for all a, b, c, d in an AG-groupoid with left identity. If for all a, b, c in an AG-groupoid G, ab = ac implies that b = c, then G is called left cancellative. Similarly, if ba = ca implies that b = c, then G is called right cancellative. It is known [5] that every left cancellative AG-groupoid is right cancellative but the converse is not true. However, every right cancellative AG-groupoid with left identity is left cancellative. Clifford and Miller [1] have defined an element zl as a left zeroid in a semigroup G if for each element x in G, there exists a in G such that ax = zl. 2000 Mathematics Subject Classification: 20N02