On prevarieties of logic

Abstract

It is proved that every prevariety of algebras is categorically equivalent to a ‘prevariety of logic’, i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in the language $$\wedge ,\vee ,\circ $$ holds in the congruence lattices of all members of every variety of logic, and that being a (pre)variety of logic is not a categorical property.