Lattices of Quasi-equational Theories as Congruence Lattices of Semilattices with Operators

We show that for every quasivariety K of relational structures there is a semilattice S with operators such that the lattice of quasiequational 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, then there is a quasivariety Q such that the lattice of theories of Q is isomorphic to Con(S,+, 0) (with no operators). We prove that if S is a semilattice having both 0 and 1 with a group G of operators acting on S, then there is a quasivariety W such that the lattice of theories of W is isomorphic to Con(S,+, 0, G). 1. Motivation and terminology Our objective is to provide, for the lattice of quasivarieties contained in a given quasivariety (Q-lattices in short), a description similar to the one that characterizes the lattice of subvarieties of a given variety as the dual of the lattice of fully invariant congruences on a countably generated free algebra. The setting for varieties is traditionally algebras, i.e., sets with operations, whereas work on quasivarieties normally allows relational structures, i.e., sets with operations and relations. The adjustments required for the more general setting are rather straightforward. In particular, we need to recall the extended definition of congruence. (See section 1.4 of Gorbunov [17]; the originals are in Gorbunov and Tumanov [19, 20] and Gorbunov [15].) A congruence on a relational structure A = 〈A,F,R〉 is a pair θ = 〈θ0, θ1〉 where • θ0 is an equivalence relation on A that is compatible with the operations of F, and • θ1 is a set of relations R ′ on A (in the language of A) such that for each relation symbol R we have R ⊆ R (i.e., the original relations of A are contained in those of θ1), and for each R ′ ∈ θ1, if R (a) holds and a θ0 b componentwise, then R (b) also holds. Date: September 1, 2007. 1991 Mathematics Subject Classification. 08C15, 08A30, 06A12.