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

by left magma-e-magma, i mean a set containing a xed element e, and equipped with the two binary operations \ and ?, with the property of e?(x y) = e?(x?y), namely the left e-join law. thus (x; ; e;?) is a left magma-e-magma if and only if (x; ) and (x;?) are magmas (groupoids), e 2 x and the left e-join law holds. the right and two-sided magma-e-magmas are de ned in an analogous way. also x is a magma-joined-magma if it is magma-x-magma for all x 2 x. therefore, i introduce a big class of basic algebraic structures with two binary operations, some of whose sub-classes are group-e-semigroups, loop-e-semigroups, semigroup-e-quasigroups and etc. a nice in nite (resp. nite) example of them is the real group-grouplike (r;+; 0;+1) (resp. klein group-grouplike). in this paper, i introduce and study the topic, construct several big classes of such algebraic structures and characterize all the identical magma-e-magmas in several ways. the motivation of this study lies in some interesting connections to f-multiplications, some basic functional equations on algebraic structures and grouplikes (recently introduced by me). finally, i present some directions for the researches conducted on the sub- ject.