Document Type : Research Paper
Islamic Azad University - Shiraz Branch
In this paper 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. Now, the researches have enabled me to
introduce Grouplikes and prove some of their main theorems
and construct a vast class of them, here. We prove
a fundamental structure theorem for a big class of grouplikes, namely
Class United Grouplikes. Moreover, we obtain some other results for
binary systems, semigroups and groups in general and exhibit several
their important subsets with related diagrams. Finally. we show some of future directions
for the researches in grouplikes and semigroup theory.