GROUPLIKES

We introduce and study an algebraic structure, namely Grouplike. A grouplike is something between semigroup and group and its axioms are generalizations of the four group axioms. Every grouplike is a semigroup containing the minimum ideal that is also a maximal subgroup (but the converse is not valid). The first idea of grouplikes comes from b-parts and b-addition of real numbers introduced by the author. Here, the aim is extend the notion of grouplikes (including sub-grouplikes, dual grouplikes, grouplike-homomorphisms with standard kernels, etc.), establish some main results and construct an expanded class. We prove a fundamental structure theorem for a large class of grouplikes, namely Class United Grouplikes. Moreover, we obtain some other results for magmas, semigroups and groups in general, exhibit several of their important subsets with related diagrams and give many equivalent conditions for semigroups to be grouplikes. Finally, we point out some directions for further research in grouplikes and semigroup theory.