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.