Universal class

Summary

A universal class is a class of structures axiomatized by universal sentences, those of the form "for all $x_1, \ldots, x_n$, a quantifier-free condition holds". Universal sentences are preserved by substructures, so every universal class is closed under substructures. By the Łoś–Tarski preservation theorem, an elementary class is universal exactly when it has this closure property. Varieties, quasivarieties and universal Horn classes are universal classes.