Defining subdirect product closed classes in infinitary logic

We note that a class of models is subdirect product closed if and only if it is definable by a class of L∞∞-sentences of a special form.


Introduction
By a weak implication we mean a sentence in L ∞∞ of the form where i for i ∈ I, and δ are atomic first-order formulas,x = (x k ) k∈K , and  [2]. We write then M SD s∈S M s . A class C of models is subdirect product closed if every subdirect product of models from C also belongs to C. (All considered classes are assumed to be classes of models in the same fixed first-order language L and are closed under taking isomorphic images.) In this note we would like to present the following observation.

Theorem. A class is subdirect product closed if and only if it is definable by a class of weak implications.
The proof of this fact is obtained by appropriate modifications in the proof of A. Shafaat's theorem [11], which states that the class of models is submodel and direct product closed (is a prevariety) if and only if it is definable by possibly infinite implications.
In Section 3, we describe an example showing that we cannot obtain an axiomatization by a set of infinite sentences of a bit more general form than weak implications even for the class P S (M) of all subdirect products of a given finite model M. This is contrary to the situation for prevarieties.
Classes typically considered in universal algebra, such as prevarieties, quasivarieties, and varieties, are subdirect product closed. But the fact that they are closed under additional operations, such as taking subalgebras, ultraproducts, and homomorphic images, makes their axiomatizations simpler. Classes of models closed solely under taking subdirect products appear in abstract algebraic logic as classes of Suszko reduced matrices for deductive systems [5], in particular as classes of reduced matrices for protoalgebraic deductive systems [4]. In the protoalgebraic case though, we also have a simpler axiomatization. Indeed, by [3, Theorem 13.10], such classes may be defined by finite implications (quasi-identities) and one possibly infinite sentence of the form where i are atomic first-order formulas and there are only countable many variables inz. However, no particular axiomatization is known in the general case.
The problem of the existence of a first-order axiomatization for P S (M), where M is finite, was considered by G. Grätzer in [7], and later revisited by J. T. Baldwin and M. A. Samhan in [1], where certain sufficient conditions were presented.
Let us recall an important result of R. C. Lyndon [10,9]. He showed that subdirect product closed classes that are first-order definable are in fact definable by first-order special Horn sentences.
Finally, note that there is an easy semantical characterization of subdirect product closed classes. For a class C and a model M let Con C (M) be the ordered set of congruences α of M such that M/α ∈ C. For algebras the order is set inclusion, and for models it is pairwise set inclusion (see [6] for the definition of congruence for models). Then C is subdirect product closed if and only if Con C (M) is a complete lattice, for every M.

Proof of the theorem
We start by verifying the easier forward direction of the Theorem.

Example
If we can define a subdirect product closed class C by weak implications in which the number of universal quantifiers is bounded by a cardinal κ, then all models of cardinality greater than κ belong to C. Thus, we cannot expect in general an axiomatization by a set of weak implications. The following example shows that it is also the case for finitely generated subdirect product closed classes even if we allow sentences of a bit more general form than weak implications.
Let ∀ κ ∃ ∞ -L ∞∞ be the language consisting of formulas in L ∞∞ of the form (∀x)(∃ȳ) γ(x,ȳ), wherex consists of less than κ variables and γ is quantifier free.   Proof. We may assume that κ is infinite. Let λ be a cardinal greater than κ.
Clearly, B is a carrier of the subalgebra B of A λ . Note that for f, g ∈ B, Then for every f ∈ B, we have f + f = f 0 and f 0 is the only idempotent element in B.
We will obtain the aim by verifying that every sentence in ∀ κ ∃ ∞ -L ∞ valid in P S (A) is also valid in B and showing that B ∈ P S (A). The first fact follows from the following claim and Lemma 3.1.

Claim.
For every subset C of B with |C| κ, there exists D B such that C ⊆ D and D ∈ P S (A).
Let I = f ∈C supp(f ). As |C| κ, so |I| κ and there exists i * ∈ λ − I. Let h be the element of B such that h(i * ) = 1 and h(i) = 0 for i = i * . Put Clearly, E B, and by (L0), D B. Observe that for a homomorphism ψ : E → A and an element a ∈ A − ψ(E), the mapping given by ϕ(g) = ψ(g) for g ∈ E and ϕ(h) = a is a homomorphism from D into A. In order to show that D ∈ P S (A), we have to find an onto homomorphism ϕ : D → A separating g 1 and g 2 for every pair g 1 , g 2 of distinct elements in D. If g 1 (i) = g 2 (i) for some i ∈ I − {0}, then as ϕ we can take the projection on the i-th coordinate. If g 1 (0) = g 2 (0) or {g 1 , g 2 } = {f 0 , h}, let ψ : E → A be the projection on the 0-th coordinate and define ϕ(g) = ψ(g) for g ∈ E and ϕ(h) = 2.
Note however that for a finite model M in a finite language, the class P S (M) is definable in L ℵ1ℵ1 by a sentence of the form (∀x)(∃ȳ)(∀z) γ(x,ȳ,z), where γ is quantifier free andx,ȳ are finite [7,Point 3.3].