International Journal of Algebra and Statistics Volume 1: 2(2012), 16–18 Published by Modern Science Publishers Available at: http://www.m-sciences.com Cancellative Left (Right) Regular Semigroups P. Sreenivasulu Reddy a , Guesh Yfter Tela a a Department of mathematics, Samara University, Semera, Afar Region, Ethiopia. (Received: 1 June 2012; Accepted: 19 June 2012) Abstract. A semigroup S is called regular semigroup if for every a ∈ S there exists x in S such that axa = a introduced by J. A. Green. In this paper, some preliminaries and basic concept of regular semigroups were presented. And proved that a cancellative semigroup S is left(right) regular semigroup if and only if it is a: (i) completely regular semigroup (ii) Clifford semigroup (iii) E-inversive semigroup (iv) 1-regular semigroup. 1. Introduction Regular semigroups were introduced by J. A. Green in his influential paper 1951 “On the structure of semigroups”. This was also the paper in which Green’s relations were introduced. The concept of regularity in a semigroup was adopted from an analogous for rings already considered by J. Von Neumann. The suggestion that the notion of regularity be applied to semigroups was first made by David Ree’s. Left (right) regularity in semigroups has long been studied. In 1954 Clifford [1] proved in his paper that semigroup is a band of groups if and only if it is both left and right regular. Kiss generalized left(right) regular elements of semigroups in 1972. Regular semigroups are easier to handle than arbitrary semigroups, but more importantly they play a paradigmatic role in semigroup as a whole. Definition 1.1. An element a of a semigroup S is said to be regular if there exist an element x in S such that a = axa. Definition 1.2. A semigroup S is said to be regular semigroup if every element of S is regular. Example 1.3. (i) Every group is regular. (ii) Every inverse semigroup is regular. Definition 1.4. An element a is said to be an E-inversive of semigroup S if there exist an element x in S such that (ax) 2 = ax and (xa) 2 = xa. Definition 1.5. A semigroup S is said to be E-inversive semigroup if every element of S is E-inversive. Example 1.6. (i) Every regular semigroup is an E-inversive semigroup. (ii) Every inverse semigroup is an E-inversive semigroup. 2010 Mathematics Subject Classification. 17. Keywords. regular semigroup; cancellative; 1-regular semigroup. Email addresses: skgm.org@gmail.com (P. Sreenivasulu Reddy), Gueshyftr2002@gmail.com (Guesh Yfter Tela)