Group-Joined-Semigroups and their structures

Every semigroup containing an ideal subgroup is called a homogroup, and it is a grouplike if and only if it has only one central idempotent. On the other hand, a class of algebraic structures covering group-$e$-semigroups $(G,\cdot,e,\odot)$ has been recently introduced. Here $(G,\cdot,e)$ is a group, $(G,\odot)$ is a semigroup and the $e$-join laws $e\odot xy=e\odot x\odot y$ and $xy\odot e=x\odot y\odot e$ hold. This paper shows close relations among these algebraic structures and proves that every group-$e$-semigroup is a group-$e$-homogroup. Also, we give some necessary and sufficient conditions for a group-$e$-semigroup to be group-$e$-grouplike. As some results of the study, we prove several characterizations of identical group-$e$-semigroups, a class of homogroups, and give several examples such as real $b$-group-grouplikes and the Klein group-grouplike.