A variety V (equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …, fn, x1, …, xm) where θ is a conjunction of equations in the function variables f1, …, fn and the individual variables x1, …, xm, if there are polynomial symbols p1, …, pn in the language of V such that ∀x1, …, xmθ(p1 …, pn, x1, …, xm) is a law of V. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.

In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

The theorem introduced here provides a sufficient condition for an axiomatic class to be elementary. It resulted from consideration of a definability problem in , but is introduced in a less specific context and accompanied by applications to .

As in [La, 1], [La, 2], a reflexive, transitive binary relation R on a nonempty set S will be called a quasi-ordering or q.o. of S. A set of pairwise incomparable elements of S is an antichain and a subset of S linearly ordered by R is a chain. A chain {Si: i ∈ ω} with i > j if and only if (siRsj ∧ ¬ sjRSi) will be called an infinite descending chain. If S has no infinite antichains nor any infinite descending chains, S is well quasi-ordered by R, and is called a well quasi-ordering or w.q.o. Any subset of a w.q.o. is a w.q.o. An element s of a q.o. S is minimal in S if ∀t ∈ S(tRs → sRt). A subset S′ of S is a foundation for S if ∀t ∈ S∃s ∈ S′(sRt).

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.

We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:

The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:

We do not know if the dual law

is true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):

(a) η (A ∨ B, H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.

(b) η(∃xA(x), H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B there exists an a ∈ D such that η(A (a), H′) = T.

Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):

Call this type of interpretation the new version of Beth semantics. We prove

Theorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.

Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:

Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schema ∀x(A ∨ B(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.