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

TL;DR

This paper extends the correspondence between lattices of quasi-equational theories and congruence lattices of semilattices with operators, specifically when the semilattice has both 0 and 1 and a group of operators fixing these elements.

Contribution

It proves that for semilattices with both 0 and 1 and a group of operators fixing them, there exists a quasivariety whose lattice of theories is isomorphic to the congruence lattice of such a semilattice.

Findings

Established isomorphism between theory lattices and congruence lattices for semilattices with operators.

Extended previous results to include semilattices with both 0 and 1 and group actions.

Demonstrated the existence of corresponding quasivarieties for these algebraic structures.

Abstract

Part I proved that for every quasivariety K of structures (which may have both operations and relations) there is a semilattice S with operators such that he lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety Q such that the lattice of theories of Q is isomorphic to Con(S,+,0). We prove that if S is a semilattice having both 0 and 1 with a group G of operators acting on S, and each operator in G fixes both 0 and 1, then there is a quasivariety W such that the lattice of quasi-equational theories of W is isomorphic to Con(S,+,0,G).