An equational basis for the variety generated by the following class of magmas.

Question

Let $(M;*)$ be a magma, and define a binary relation $R$ on $M$ by saying that $aRb$ iff there exists a $c$ in $M$ such that $a*c=b$. I call $R$ the left-divisor relation associated with $(M;*)$. I call a magma $(M;*)$ left-reflexive if the left-divisor relation associated with $(M;*)$ is a reflexive binary relation. I have two questions. First, is the class of left-reflexive magmas a variety (in the sense of universal algebra)? Second, whether or not it is a variety, is there a finite equational basis for the variety generated by left-reflexive magmas, and if so, can someone give me an explicit finite basis? For the second question, I conjecture that $\{x=x\}$ is a basis. That is, I conjecture that the variety generated by this class is the full variety of all magmas. Of course, one could ask analogous questions for classes of magmas whose left-divisor relation is transitive, or symmetric, or anti-symmetric, etc. I might decide to ask more questions like that in the future.

Answer 1

Given any magma $M$, consider the new magma $M^\star$ gotten from $M$ as follows:

• The domain of $M^\star$ is that of $M$, together with a new element $\star$.

• The multiplication of $M^\star$ extends that of $M$ by setting $\star \star=\star$ and $a\star=\star a=a$ for all $a\in M$.

The new magma $M^\star$ is trivially left-reflexive. Consequently:

Every magma is a submagma of a left-reflexive magma.

Since varieties are closed under taking substructures, this means that the variety generated by the class of left-reflexive magmas is the variety of all magmas.

Note that similar arguments apply to variations of this question. For example, let $M'$ be the magma with underling set that of $M$ together with a new element $c_{a,b}$ for each $a,b\in M$ as well as a last new element $\#$, with multiplication extending that of $M$, satisfying $ac_{a,b}=b$ for all $a,b\in M$, and with all results of products not determined by these rules equalling $\#$. Then the left-divisor relation of $M'$ is an equivalence relation (indeed it's all of $M'\times M'$). And so forth.