By **algebraic theory** I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe universal algebraists call these abstract clones?)

A **strict terminal object** in a category is a terminal object $1$ such that every morphism with domain $1$ is an isomorphism. A well-known example of a category with strictly terminal objects is the category of (unital) rings: the only ring homomorphisms with domain $\{ 0 \}$ are isomorphisms.

It is a fact that any model of an algebraic theory whose underlying set is a singleton must be a terminal object in the corresponding category of models. My question is,

When are terminal objects in the category of models for an algebraic theory strict?

It is easy to generalise the proof that the algebraic theory of rings has the strict terminal object property: indeed, for any algebraic theory with two constants $0$ and $1$ and a binary operation for which $0$ is a (left) absorbing element and $1$ is a (left) unit, the trivial model is strictly terminal.

Another easy observation is that any algebraic theory that "interprets" an algebraic theory with the strict terminal object property must itself have the strict terminal object property. (Here, $T$ interprets $T'$ if there is a map from the operations of $T'$ to the operations of $T$ that preserves the multi-composition and projections.) So a natural follow-up question is,

Is there a "universal" algebraic theory with the strict terminal object property?