LATTICES OF QUASI-EQUATIONAL THEORIES AS CONGRUENCE LATTICES OF SEMILATTICES WITH OPERATORS: PART II
Kira Adaricheva, James B. Nation ยท International Journal of Algebra and Computation ยท 2012
Part I proved that for every quasivariety ๐ฆ 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 ๐ฆ (the dual of the lattice of sub-quasivarieties of ๐ฆ) is isomorphic to Con(S, +, 0, ๐ก). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety ๐ฌ such that the lattice of theories of ๐ฌ is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group ๐ข of operators acting on S, and each operator in ๐ข fixes both 0 and 1, then there is a quasivariety ๐ฒ such that the lattice of theories of ๐ฒ is isomorphic to Con(S, +, 0, ๐ข).