Author/Authors :
Hooshmand، M. H نويسنده Young Researchers and Elite Club, Shiraz Branch, Islamic Azad University, Shiraz, Iran. ,
Abstract :
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?(xy) = 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.
