A Study of Regular Grouplikes

Recently, a new algebraic structure, namely grouplike, has been introduced and studied by the first author. A grouplike is something between semigroup and group with so close relations to groups. The grouplike axioms are generalizations of the four group axioms. In this paper, we study regular and irregular grouplikes as a class of semigroups. The motivation of this study lies in some interesting properties of regular proper grouplikes.