The Existence of Finitely Based Lower Covers for Finitely Based Equational Theories

By an equational theory we mean a set of equations from some fixed language which is closed with respect to logical consequences. We regard equations as universal sentences whose quantifierfree parts are equations between terms. In our notation, we suppress the universal quantifiers. Once a language has been fixed, the collection of all equational theories for that language is a lattice ordered by set inclusion. The meet in this lattice is simply intersection; the join of a collection of equational theories is the equational theory axiomatized by the union of the collection. In this paper we prove, for languages with only finitely many fundamental operation symbols, that any nontrivial finitely axiomatizable equational theory covers some other finitely axiomatizable equational theory. In fact, our result is a little more general. There is an extensive literature concerning lattices of equational theories. These lattices are always algebraic. Compact elements of these lattices are the finitely axiomatizable equational theories. We also call them finitely based. The largest element in the lattice is compact; it is the equational theory based on the single equation x ≈ y. The smallest element of the lattice is the trivial theory consisting of tautological equations. For all but simplest languages, the lattice of equational theories is intricate. R. McKenzie in [6] was able to prove in essence that the underlying language can be recovered from the isomorphism type of this lattice. A key to McKenzie’s main result involved understanding first order definability within this lattice. Our knowledge of definability in this lattice was substantially advanced in the work of Ježek [3]. Given an equational theory T , by LT we denote the sublattice of the lattice of equational theories of the given language comprised of all equational theories which include T . Thus if T were the equational theory of semigroups, then LT would be the lattice of all equational theories of semigroups and one of the members of LT would be the equational theory of commutative semigroups. Almost any algebraic lattice appears as an interval in the lattice of all equational theories, subject to the obvious cardinality restrictions, see Ježek [2] and Pigozzi and Tardos [17]. It is nevertheless true that the lattices of the form LT have not yet been clearly understood. Lampe in [4] found a series of first order conditions, each actually a universal Horn sentence, that each of the lattices LT must satisfy. As a result, even a simple lattice of five elements can be shown not representable in the form LT . For the basics of equational logic the reader is referred to either McNulty [11] or Taylor [18], and for the basics about varieties of algebras, to [8]. We are going to investigate the existence of covering in the lattice LT , where T is a term finite equational theory. We say that an equational theory T is term finite provided that for each term t, the set {s : t ≈ s ∈ T} is finite. Theorem 6, which is our main result, asserts that for every term finite equational theory T of a finite language, every equational theory properly extending T and finitely based relative to T has a lower cover extending T , which is again finitely based relative to T . In particular, if T is the trivial theory, this means that every nontrivial finitely based equational theory of a finite language has a finitely based lower cover in the lattice of equational theories. The construction is not effective; however, we can effectively construct a finite base for an equational theory E, properly contained in E and such that the number of the equational theories between E and E is finite and can be effectively estimated by an upper bound. Our proof takes much from McKenzie [6], where among other things he proved that any nontrivial equational theory has a lower cover. In McNulty [9] it was proven that if T is an