f-Grouplikes

Abstract

Grouplikes have been introduced and studied by the first author.A grouplike is something between semigroup and group and its axioms are generalizationof the four group axioms. We observe that every grouplike is a homogroup (a semigroupcontaining an ideal subgroup) with a unique central idempotent.On the other hand, decomposer and associative functions on groups, semigroupsand even magmas are introduced in 2007.If $(G,\cdot)$ is a group and $f:G\rightarrow G$ is an associative function (i.e.$f(xf(yz))=f(f(xy)z)$, for all $x,y,z\in G$), then the $f$-multiplication "$\cdot_f$"defined by $x\cdot_f y =f(xy)$, is an associative binary operation with severalinteresting properties. A nice example for associative function,$f$-multiplication and such algebraic structures are $b$-decimal part functions$(\; )_b$, $b$-addition $+_b$, and the real $b$-grouplike $(\mathbb{R},+_b)$.In this paper, we introduce an important type of grouplikes (namely $f$-grouplike) that is motivated fromthe both topics. We prove that $f$-grouplikes is a proper subclass of Class United Grouplikes, study some of their properties and show some of future directionsfor the researches.