Abstract
By left magma-$e$-magma, I mean a set containingthe fixed element $e$, and equipped by two binary operations $cdot$, $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namelyleft $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magmaif 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 basicalgebraic structures with two binary operations which some of theirsub-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 characterizeall 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 directionsfor the researches.
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