On Group-like Properties of Left Groups

A left group is a semigroup that is the direct product of a left zero semigroup and a group. In this paper, we investigate some group-like properties of left groups. In particular, we characterise monogenic left group monoids and prove some fundamental results on left group morphisms. We also characterise Green’s equivalences on left groups and, analogously to groups, establish a correspondence between congruences and normal sub-left groups. Finally, we characterise DSC left groups.