Universal Horn class

Remarks

Equivalently, each axiom has one of the forms $\forall \bar x \, (\alpha_1 \wedge \cdots \wedge \alpha_n \rightarrow \beta)$ or $\forall \bar x \, \neg (\alpha_1 \wedge \cdots \wedge \alpha_n)$ with atomic $\alpha_i$ and $\beta$. Universal Horn sentences are preserved by substructures and by direct products of nonempty families, so every universal Horn class is closed under these constructions.

Example: partial orders, in the signature with one binary relation $\le$, form a universal Horn class. Its axioms $\forall x \, (x \le x)$, $\forall x \forall y \, (x \le y \wedge y \le x \rightarrow x = y)$ and $\forall x \forall y \forall z \, (x \le y \wedge y \le z \rightarrow x \le z)$ are universal Horn sentences.

Non-example: total orders do not form a universal Horn class. The product of the 2-element chain $\set{0 < 1}$ with itself, ordered coordinatewise, is a partial order in which $(0, 1)$ and $(1, 0)$ are incomparable, so the class is not closed under products.

When $\sigma$ has only function symbols, the universal Horn sentences of the first form are exactly the quasi-identities, so the corresponding classes are the quasivarieties.