Hooshmand, M. (2015). MAGMA-JOINED-MAGMAS: A CLASS OF NEW ALGEBRAIC STRUCTURES. Journal of Algebraic Systems, 3(2), 171-199. doi: 10.22044/jas.2015.616

M. H. Hooshmand. "MAGMA-JOINED-MAGMAS: A CLASS OF NEW ALGEBRAIC STRUCTURES". Journal of Algebraic Systems, 3, 2, 2015, 171-199. doi: 10.22044/jas.2015.616

Hooshmand, M. (2015). 'MAGMA-JOINED-MAGMAS: A CLASS OF NEW ALGEBRAIC STRUCTURES', Journal of Algebraic Systems, 3(2), pp. 171-199. doi: 10.22044/jas.2015.616

Hooshmand, M. MAGMA-JOINED-MAGMAS: A CLASS OF NEW ALGEBRAIC STRUCTURES. Journal of Algebraic Systems, 2015; 3(2): 171-199. doi: 10.22044/jas.2015.616

MAGMA-JOINED-MAGMAS: A CLASS OF NEW ALGEBRAIC STRUCTURES

^{}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.