Axiomatizations of universal classes through inﬁnitary logic

. We present a scheme for providing axiomatizations of universal classes. We use inﬁnitary sentences there. New proofs of Birkhoﬀ’s HSP theorem and Mal’cev’s SPP U -theorem are derived. In total, we present 75 facts of this sort.


Unary class operators
In this paper we present an easy and unified way for proving preservation theorems for various types of universal classes. We start with a basic observation. It is a formalization of a trick which may be encountered in various proofs of preservation theorems. In order to do so, we first need to clarify the notion of unary class operator.
Let L be a fixed first-order language, i.e., a set of functional and relational symbols of finite arities. Let Struc(L) be the class of all structures in L (called also L-structures). By a unary class operator we mean an assignment O which with each structure A in Struc(L) associate a certain class O(A) ⊆ Struc(L) such that the following two requirements are fulfilled (The question whether such objects exist is a set theoretical issue. However, as it is commonly done in set theory (see e.g. [12,Chapter 8]), we may assume that there is a first-order formula Φ(x, y) in the language of set theory such that O(A) = { B | B ∈ Struc(L) and Φ(A, B)}, and then identify O with Φ.) Note that we do not impose the idempotency condition OO = O. We may compose unary class operators. So C ∈ OO (A) if there exists B such that C ∈ O(B) and B ∈ O (A).
In this paper we deal only with two class operators which are not unary: the direct product P and the ultraproduct P U class operators. Thus we do not need to define this notion. We do it just for the sake of completeness. Let O be a unary class operator. Assume that with each structure A ∈ Struc(L) we may associate a sentences χ O A , say in L ∞,∞ (see e.g. [3] for definitions in infinitary logic), such that

Classical applications
By a disjunctive universal sentence we mean a sentence in L ∞,∞ of the form where all formulas ϕ i and ψ j are atomic and all its variables are in X. The subformula i∈I ¬ϕ i is called the negative part of σ and J∈J ψ j is called the Thus, if the positive part consists of one atomic formula, then we speak about implications. A first-order implication, i.e., in L ω,ω , is called a quasi-identity.
And an identity is a positive disjunctive universal sentence with exactly one disjunct.
A general scheme of theorems in this paper is as follows: A class of Lstructures is closed under a particular class operator O if and only if it is definable by sentences of a special form. The proofs split into two parts. We should check that if σ is of the special form, A |= σ and B ∈ O(A), then B |= σ. Secondly, we find characteristic sentences for O which are of the considered form. Usually, the first part is easy. Hence, in most cases we just argue for the second part.
Let A be an L-structure and A be its carrier. Let X A be a set of variables for which there is a bijection π A : A → X A . For a set X of variables let At(X) be the set of atomic formulas with all variables in X. Define Then we have the following facts. We immediately obtain the following fact. We are mainly interested in first-order axiomatizations. We use the following known facts. (For (1) see e.g. [13,Theorem 7.5.2]. Point (2) may be proved using the same construction as in Mal'cev's proof of compactness theorem [13, theorem 8.3.1].) Recall that P denotes the direct product and P U the ultraproduct class operators. Lemma 2.3. Let σ = ∀X i∈I ϕ i → j∈J ψ j be a disjunctive universal sentence. Assume that the class C satisfies σ.  (1) and (2).
The above proof is non-constructive. The original proof of Birkhoff's theorem has a different character. It is longer, but also has an advantage that it connects equational theories with free algebras. Similarly for Mal'cev's theorem, there is a connection of a quasi-equational theories and finitely presented algebras. It is used in categorical generalizations of Mal'cev's theorem [1, Section 16]. Although proofs of Mal'cev's theorem presented in most textbooks are, as our, non-constructive.

Other applications
It appears that the closure under other unary class operators leads to various restrictions on the defining sentences.

Restrictions on the negative part
In the previous section we had an extreme situation. The unary class operator HS completely eliminated the negative part. But we may consider weaker restrictions: forbidding occurrences of the equality symbol ≈ or occurrences of relational symbols different than ≈, called from now on just relational symbols. This may be achieved by considering various types of homomorphisms. Let L R be the set of relational symbols in L (i.e., non-functional symbol different than ≈). For A ∈ Struc(L) and R ∈ L R let R A denotes the interpretation of R in A.
We say that A is a relational expansion of B if A and B have the same algebraic reduct and R B ⊆ R A for every R ∈ L R . Thus, informally speaking, A is obtained from B by adding new tuples of elements from A to relations. Let H E (A) be the class of structures isomorphic to relational expansions of A.
We say that a homomorphism h : Strict homomorphisms appear naturally in abstract algebraic logic [7], see also [8]. Let H Str (A) be the class of all strict homomorphic images of A. Proof. The reasoning here is the same as in the proof of Theorem 2.2. We only provide characteristic sentences. So the sentences ψ has no occurrences of relational symbols ψ are characteristic for H E S. Note that χ H E S may be reduced by adding the condition in the first big conjunct that ϕ has no occurrences of functional symbols. Further, the sentences In what follows, every theorem would be accompanied by a statement like Corollary 3.2. They are derivable in a straightforward way. Thus we omit them.
There is a type of homomorphism commonly occurring in the literature, mostly implicitly, with the connection to congruences. We say that a homo- precisely, if B |= R(b 0 , . . . , b n−1 ), then there are a 0 , . . . , a n−1 ∈ A such that A |= R(a 0 , . . . , a n−1 ) and h(a i ) = b i for all i < n. Informally, if there is a tuple in a relation of B, then there is a witness of it in A. Note that if γ is a congruence on the algebraic reduct of A, then there is a unique structure A/γ such that a → a/γ is a strong homomorphism from A onto A/γ. On the other side, if h : A → B is a surjective strong homomorphism, then B is isomorphic to A/γ, where γ is the kernel of h. Let H Sng (A) be the class of all strong homomorphic images of A.
It appears that the closure under H Sng is very restrictive. The following fact was observed in [2, Section 5, Point (3)]. Our proof falls into the presented scheme. We say that a disjunctive universal sentence is weak if in its negative part every variable appears at most once and there are no occurrences of ≈ nor of functional symbols.
. Indeed, we may define μ as follows. Let X i be the set of variables occurring in ϕ i . Firstly, we define μ i : X i → A. Assume that ϕ i = R(x 0 , . . . , x n−1 ). Then X i = {x 0 , . . . , x n−1 }. Since B |= R(ν(x 0 ), . . . , ν(x n−1 )) and h is strong, there are a 0 , . . . , a n−1 ∈ A such that A |= R(a 0 , . . . , a n−1 ) and h(a i ) = ν(x i ) for i < n. We put μ i (x i ) = a i . Since variables x 0 , . . . , x n−1 are mutually distinct, the definition of μ i is correct. Furthermore, since X i ∩ X i = ∅ for i = i , we may define μ(x) = μ i (x) when x ∈ X i for some i ∈ I. Finally, if x is a variable which does not appear in the negative part of σ, then as μ(x) we take any element from h −1 (ν(x)).
Let us argue that the closure under H Sng S is a restrictive condition. Assume that our fixed language L is finite and has only relational symbols. Then there are only finitely many subvarieties (HSP-closed classes, i.e., classes defined by identities) of Struc(L). Indeed, there are, up to logical equivalence, only finitely many identities in L. Hence considering varieties for relational structures is not interesting. What we gain when we shift to H Sng SP-closed classes? Actually, not too much. There are still only finitely many such classes.
In order to see this let us recall the notion of relative subdirect irreducibility. Let C be a S-closed subclass of Struc(L) and S be a structure in C. We say that S is C-subdirectly irreducible if for every embedding e : S → i∈I A i , where A i ∈ C, there is a projection π k : i∈I A i → A k such that the composition π k • e : S → A k is an embedding. We say that S is H Sng -subdirectly irreducible if S is H Sng SP(S)-subdirectly irreducible. Then S is C-subdirectly irreducible for some H Sng SP-closed class C iff it is H Sng -subdirectly irreducible. Recall also a known fact that if C is a quasivariety (SPP U -closed class) then every algebra in C embeds into a product of C-subdirectly irreducible algebras [10, Theorem 3.1.1]. Every H Sng and P-closed class is P U -closed. Hence every H Sng SP-closed class C is a quasivariety and the recalled fact may be applied to C. We thus infer that every structure embeds into a product of H Sng -subdirectly irreducible structures.
The next proposition shows that if L is finite and has no functional symbols, then there are, up to isomorphism, only finitely many H Sng -subdirectly irreducible L-structures. Consequently, by the conclusion from the previous paragraph, there are only finitely many H Sng SP-closed classes for such L. Proof. Let A be a structure in L. Then every equivalence relation γ on A induces the strong homomorphism η γ : A → A/γ; a → a/γ. For every pair a, b of distinct elements of A let γ a,b be an equivalence relation on A such that (a, b) ∈ α and |A/α| = 2. For every R ∈ L R of arity n and every tuplē a = (a 0 , . . . , a n−1 ) such that A |= R(a 0 , . . . , a n−1 ) let γ R,ā be the equivalence relation given by (A − {a 0 , . . . , a n−1 }) 2 ∪ {(a 0 , a 0 ), . . . , (a n−1 , a n−1 )}. Then |A/γ R,ā | ≤ n + 1. We have   A/γ a,b |= a/γ a,b ≈ b/γ a,b and A/γ R,ā |= ¬R(a 0 /γ R,ā , . . . , a n−1 /γ R,ā ).

This yields that A embeds into the product
of structures each of which is a strong homomorphic image of A and has the carrier of size at most max(m + 1, 2). However, it is not difficult to see that there are denumerably many H Sng SP U -closed classes if the language has no functional neither relational symbols. Indeed for every n > 0 the sentence expresses "there are at most n elements". Thus they are mutually logically nonequivalent. Also, there are continuum many quasivarieties generated by simple graphs [5, Theorem 2].  {(a, d), (d, a), (b, c), (c, b)} ∪ D, where D is the diagonal in A 2 . Then the strong homomorphic image A/γ is not transitive and A/δ is not antisymmetric.
We have the same situation in case of ordered algebras. Let us consider ordered semigroups. Let N = (N, +, ≤) be the structure of natural numbers with the standard addition and order. Then for a congruence γ on (N, +) the strong homomorphic image N/γ is transitive but it does not have to be antisymmetric. Indeed, it is antisymmetric iff γ has at most one nontrivial class. Also transitivity is not preserved by strong homomorphisms for ordered semigroup. Let S = (N 4 , +, ) be a structure, where + is the standard componentwise addition and be the order is given in the following way: (r 0 , r 1 , r 2 , r 3 ) (s 0 , s 1 , s 2 , s 3 ) iff r 0 + r 1 = s 0 + s 1 , r 2 + r 3 = s 2 + s 3 and r 1 ≤ s 1 , r 3 ≤ s 3 . It is a least order on N 4 which is compatible with the addition and such that e 0 ≤ e 1 , e 2 ≤ e 3 , where e i is the quadruple with three 0s and with 1 in the i'coordinate. Let γ be the congruence on (N 4 , +) generated by (e 1 , e 2 ). Then in S/γ we have e 0 /γ e 1 /γ = e 2 /γ e 3 /γ. But, since e 0 /γ = {e 0 }, e 3 /γ = {e 3 } and not e 0 e 3 , we do not have e 0 /γ e 3 /γ.
The restrictive character of H Sng S leads us to the following question. Is there a notion weaker than strict homomorphism and stronger than strong homomorphism which corresponds to disjunctive universal sentences without occurrences of ≈ nor of functional symbols in the negative part but allowing

Restrictions on the positive part and on both parts
Note that the situation is not symmetric in the following sense. For every disjunctive universal sentence there is a logically equivalent disjunctive universal sentence without functional symbols and repetitions of variables in the positive part and without equations of the form x ≈ y in the negative part. Thus, considering restrictions on the positive part, we only provide analogs of Theorems 2.2 and 3.1.
If O is a unary class operator, then O −1 is the unary class operator given by: Proof. We provide characteristic sentences for Points (2) and (3). So the sentences ¬ R(π A (a 0 ), . . . , π A (a n−1 )) ∧ With every pair of unary class operators O 1 , O 2 in Theorem 3.8 in the table there are connected additional statements analogical to Corollary 3.2 which concern the closure under P and P U . Thus in total, we obtain (4 × 6 − 1) × 4 − 17 = 75 statements. We subtract 1 since the case when O 1 = O 2 = H is trivial (we obtain the class of all L-structures). We subtract 17 since whenever a class is H Sng and P-closed, then it is P U -closed, and whenever it is H −1 -closed, then it is P-closed (for example a H Sng H −1 -closed class is PP U -closed).
The proofs of statements which where not considered earlier fall into the same schema but have more complicated details. Let us finish the paper with an example of such reasoning.
Proof. (Sample proof) We argue for the HH −1 Str S unary class operator. We check that the sentences For the inverse implication, assume that ν : X A → B is a valuation such that B |= ϕ[ν] for every ϕ ∈ Diag − (A) without occurrences of ≈. We want to find homomorphisms h : C → A and g : C → B such that h is surjective and g is strict. Let the algebraic reduct C alg of C be the algebra of terms with variables in X A . Let h : C alg → A alg be a unique algebraic homomorphism extending (π A ) −1 . Similarly, let g : C alg → B alg be a unique algebraic homomorphism extending ν. We uniquely define the relational reduct of C in such a way that g is a strict homomorphism. Then, by assumption on χ HH −1 Str S A , h is a homomorphism. For the H Str H −1 Str S-case one may also consult [8, Lemma 5].