LATTICES OF QUASI-EQUATIONAL THEORIES AS CONGRUENCE LATTICES OF SEMILATTICES WITH OPERATORS: PART II

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 the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡). 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 theories of 𝒲 is isomorphic to Con(S, +, 0, 𝒢).