Document Type: Original Manuscript

Author

Young Researchers and Elite Club, Shiraz Branch, Islamic Azad University, Shiraz, Iran.

Abstract

By left magma-$e$-magma, I mean a set containing
the fixed element $e$, and equipped by two binary operations "$cdot$"
, $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namely
left $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magma
if and only if $(X,cdot)$, $(X,odot)$ are magmas (groupoids), $ein X$ and the left $e$-join law holds.
Right (and two-sided) magma-$e$-magmas are defined in an analogous way.
Also, $X$ is magma-joined-magma if it is magma-$x$-magma, for all $xin X$. Therefore, we introduce a big class of basic
algebraic structures with two binary operations which some of their
sub-classes are group-$e$-semigroups, loop-$e$-semigroups, semigroup-$e$-quasigroups,
etc. A nice infinite [resp. finite] example for them is real group-grouplike $(mathbb{R},+,0,+_1)$ [resp. Klein group-grouplike].
In this paper, I introduce and study the topic, construct several big classes of such algebraic structures and characterize
all identical magma-$e$-magma 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 been introduced by the author). At last, we show some of future directions
for the researches.

Keywords