magma-joined-magmas: a class of new algebraic structures

By left magma-e-magma, i mean a set containing axed element e, and equipped with the two binary operations
and ?, with the property of e?(x
y) = e?(x?y), namely the lefte-join law. thus (x;
; e;?) is a left magma-e-magma if and onlyif (x;
) and (x;?) are magmas (groupoids), e 2 x and the lefte-join law holds. the right and two-sided magma-e-magmas aredened in an analogous way. also x is a magma-joined-magma ifit is magma-x-magma for all x 2 x. therefore, i introduce a bigclass of basic algebraic structures with two binary operations, someof whose sub-classes are group-e-semigroups, loop-e-semigroups,semigroup-e-quasigroups and etc. a nice innite (resp. nite)example of them is the real group-grouplike (r;+; 0;+1) (resp.klein group-grouplike). in this paper, i introduce and study thetopic, construct several big classes of such algebraic structures andcharacterize all the identical magma-e-magmas in several ways.the motivation of this study lies in some interesting connectionsto f-multiplications, some basic functional equations on algebraicstructures and grouplikes (recently introduced by me). finally, ipresent some directions for the researches conducted on the sub-ject.