What is the name of this kind of algebra?

Question

Consider a set $S$ with a single binary operation $*$. Suppose that the only equations $*$ satisfies are of the form $t=t$, for every term $t$. What would be the name of this kind of algebra? Of course, the signature doesn't have to consist of a single binary operation. It could be any number of operations with any number of arities. But the essential thing is, they only satisfy equations of the form $t=t$. Is there a standard name for this kind of algebra?

Keith Kearnes · Answer

I will assume that when you say equation you mean universally quantified equation, or identity.
If $mathcal V$ is a variety, then an algebra that generates $mathcal V$ is called generic for $mathcal V$. Free algebras on sufficiently many generators are generic, but generic algebras need not be free.
This applies here. Suppose that $langle S; *rangle$ only satisfies trivial identities. Then $S$ is generic for the variety of algebras with a single binary operation. But it need not be free. For example, $S' = Stimes T$ where $T$ is a $2$-element semigroup will satisfy the same identities as $S$ (only trivial ones), but will not be free.

What is the name of this kind of algebra?

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