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](http://math.stackexchange.com/questions/865476/what-do-we-call-those-functions-that-can-be-obtained-from-term-operations-by-par) is a related terminological question I asked the other day.