If an algebra of type $A$ contains a subalgebra which is also an algebra of type $B$, then it is a Smarandache $B$-type $A$-algebra provided the subalgebra of type $B$ contains at least two elements. This generalizes the notion of Smarandache group, where the group of order $\geq 2$ is a subsemigroup of a semigroup. In this paper we investigate a number of such pairings and we deduce a number of conclusions as a consequence.

In this paper, we investigate the effects of certain variants of the commutative laws on properties of several families of algebras which are in general not commutative, such as groups, linear algebras, or actually quite anti-commutative, such as $BCK$-algebras and $d$-algebras among others. From results obtained it becomes clear that by considering these variants in the presence of yet other axioms it is to be expected that a quite rich and varied set of results may be obtained both in the general and the particular setting of which what has been accomplished in this paper is a substantial sample.

In this paper, we observe that if X is a set and (Bin(X), □) is the semigroup of binary systems on X, then its center ZBin(X) consists of the locally-zero-semigroups and that these can be modeled as (simple) graphs and conversely. Using this device we show that we may obtain many results of interest concerning groupoids by reinterpreting graph theoretical properties and at the same time results on graphs G may be obtained by considering them as elements of centers of the semigroups of binary systems (Bin(X), □) where X = V(G), the vertex set of G.