Summary: By left magma-\(e\)-magma, I mean a set containing the fixed element \(e\), and equipped by two binary operations ``\(\cdot\)'' and \(\odot\) with the property \(e\odot (x\cdot y)=e\odot(x\odot y)\), 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), \(e\in X\) 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 \(x\in X\). 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 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 been introduced by the author). Finally, I present some directions for the researches conducted on the subject.