Lattices of Theories in Languages without Equality

If S is a semilattice with operators, then there is an implicational theory Q such that the congruence lattice Con(S) is isomorphic to the lattice of all implicational theories containing Q. The author and Kira Adaricheva have shown that lattices of quasi-equational theories are isomorphic to congruence lattices of semilattices with operators [1]. That is, given a quasi-equational theory Q, there is a semilattice with operators S such that the lattice QuTh(Q) of quasi-equational theories containing Q is isomorphic to Con(S). There is a partial converse: if the semilattice has a largest element 1, and under strong restrictions on the monoid of operators, then Con(S,+, 0,F) can be represented as a lattice of quasiequational theories. Any formulation of a converse will necessarily involve some restrictions, as there are semilattices with operators whose congruence lattice cannot be represented as a lattice of quasi-equational theories. In particular, one must deal with the element corresponding to the relative variety x ≈ y, which has no apparent analogue in congruence lattices of semilattices with operators. In this note, it is shown that if S is a semilattice with operators, then Con(S,+, 0,F) is isomorphic to a lattice of implicational theories in a language that may not contain equality. The proof is a modification of the previous argument [1], but not an entirely straightforward one. En route, we also investigate atomic theories, the analogue of equational theories for a language without equality. For classical logic without equality, see Church [4] or Monk [12]. More recent work includes Blok and Pigozzi [2], Czelakowski [5], and Elgueta [6]. The standard reference for quasivarieties is Viktor Gorbunov’s book [7]. The rules for deduction in implicational theories are given explicitly in section 3. Our main result, Corollary 8, of course depends on these. It does not depend on the model theory used to interpret how it applies to structures, and indeed there are options in this regard. So there are two Date: November 8, 2010. 1991 Mathematics Subject Classification. 08C15, 08B15, 03G10.