Let $T$ denote an algebraic theory.

Terminological Question. Let $X$ denote a $T$-algebra. Is there a name for the preorder $\mid$ defined on $X$ by asserting that $a \mid b$ iff there is a term operation $f : X^n \times X \rightarrow X$ such that $f(\tilde{x},a)=b$ for some $\tilde{x} \in X^n$?

Even if no such name exists, I am interested to read more about this relation. In particular, I'd like to know:

Main Question. For which algebraic theories $T$ does it hold that the $\mid$ preorder (as defined above) is antisymmetric on all free $T$-algebras?

Examples/counterexamples.

• Let $T$ denote the theory of Abelian monoids. Then every free $T$-algebra has the property of interest.
• Let $T$ denote the theory of Abelian groups. Then no non-trivial $T$-algebra has the property of interest.

Here is a related terminological question I asked the other day.

Let $T$ denote an algebraic theory.

Terminological Question. Let $X$ denote a $T$-algebra. Is there a name for the preorder $\mid$ defined on $X$ by asserting that $a \mid b$ iff there is a term operation $f : X^n \times X \rightarrow X$ such that $f(\tilde{x},a)=b$ for some $\tilde{x} \in X^n$?

Even if no such name exists, I am interested to read more about this relation. In particular, I'd like to know:

Main Question. For which algebraic theories $T$ does it hold that the $\mid$ preorder (as defined above) is antisymmetric on all free $T$-algebras?

Examples/counterexamples.

• Let $T$ denote the theory of Abelian monoids. Then every free $T$-algebra has the property of interest.
• Let $T$ denote the theory of Abelian groups. Then no non-trivial $T$-algebra has the property of interest.

Here is a related terminological question I asked the other day.