Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I

TL;DR

This paper demonstrates that the lattice of quasi-equational theories for any quasivariety can be represented as the congruence lattice of a semilattice with operators, revealing new structural restrictions.

Contribution

It establishes an isomorphism between the lattice of quasi-equational theories and congruence lattices of semilattices with operators, providing a new structural perspective.

Findings

Lattices of quasi-equational theories are isomorphic to congruence lattices of semilattices with operators.

New restrictions on the natural quasi-interior operator are identified.

The dual of the lattice of sub-quasivarieties is characterized through this isomorphism.

Abstract

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.