# On prevarieties of logic

## TL;DR

This paper demonstrates that every prevariety of algebras can be represented as a prevariety of logic, revealing insights into the algebraic semantics of deductive systems and properties of congruence lattices.

## Contribution

It establishes the categorical equivalence between prevarieties of algebras and prevarieties of logic, and analyzes properties of congruence lattices in logical varieties.

## Key findings

- Every prevariety of algebras is categorically equivalent to a prevariety of logic.
- No nontrivial equation in meet, join, and relational product holds universally in congruence lattices of logical varieties.
- Being a (pre)variety of logic is not a categorical property.

## 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 "meet, join, and relational product" 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.