Elementary class

Summary

An elementary class is the class of all models of a first-order theory in a fixed signature. Groups, rings, fields of characteristic $0$ and dense linear orders are elementary classes. An elementary class is closed under elementary equivalence and under ultraproducts. By the compactness theorem, many natural classes, such as torsion groups, are not elementary. Universal classes, universal Horn classes, quasivarieties and varieties are elementary classes with axioms of special shapes.