Cancellative Left (Right) Regular Semigroups

A semigroup $S$ is called regular semigroup if for every $a \in S$ 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) $g$-regular semigroup.