All congruence lattice identities implying modularity have

For an arbitrary lattice identity implying modularity (or at least congruence modularity) a Mal’tsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T034137 and T026243, and also by the Hungarian Ministry of Education, grant no. FKFP 0169/2001. It is an old problem if all congruence lattice identities are equivalent to Mal’tsev (=Mal’cev) conditions. In other words, we say that a lattice identity λ can be characterized by a Mal’tsev condition if there exists a Mal’tsev condition M such that, for any variety V, λ holds in congruence lattices of all algebras in V if and only if M holds in V; and the problem is if all lattice identities can be characterized this way. This problem was raised first in Grätzer [15], where the notion of a Mal’tsev condition was defined. A strong Mal’tsev condition for varieties is a condition of the form ”there exist terms h0, . . . , hk satisfying a set Σ of identities” where k is fixed and the form of Σ is independent of the type of algebras considered. By a Mal’tsev condition we mean a condition of the form ”there exists a natural number n such that Pn holds” where the Pn are strong Mal’tsev conditions and Pn implies Pn+1 for every n. The condition ”Pn implies Pn+1” is usually expressed by saying that a Mal’tsev condition must be weakening in its parameter. (For a more precise definition of Mal’tsev conditions cf. Taylor [23].) The problem was repeatedly asked by several authors, including Taylor [23], Jónsson [13] and Freese and McKenzie [11]. Certain lattice identities have known characterizations by Mal’tsev conditions. The first two results of this kind are Jónsson’s characterization of (congruence) distributivity by the existence of Jónsson terms, cf. Jónsson [12], and Day’s characterization of (congruence) modularity by the existence of Day terms, cf. Day [8]. Since Day’s result will be needed in the sequel, we formulate it now. For n ≥ 2 let (Dn) denote the strong Mal’tsev condition ”there are quaternary terms m0, . . . , mn satisfying the identities m0(x, y, z, u) = x, mn(x, y, z, u) = u, mi(x, y, y, x) = x for i = 0, 1, . . . , n, mi(x, x, y, y) = mi+1(x, x, y, y) for i = 0, 1, . . . , n, i even, mi(x, y, y, z) = mi+1(x, y, y, z) for i = 0, 1, . . . , n, i odd”. Now Day’s celebrated result says that a variety V is congruence modular iff the Mal’tsev condition ”(∃n)(Dn)” holds in V. Jónsson terms and Day terms were soon followed by some similar characterizations for other lattice identities, given for example by Gedeonová [14] and Mederly [19], but Nation [20] and Day [9] showed that these Mal’tsev conditions are equivalent to the existence of Day terms or Jónsson terms; the reader is referred to Jónsson [13] and Freese and McKenzie [11, Chapter XIII] for more details. The next milestone is Chapter XIII in Freese and McKenzie’s book [11]. Let us call a lattice identity λ in n variables a frame identity if λ implies modularity and λ holds in a modular lattice iff it holds for the elements of every (von Neumann) n-frame of the lattice. Freese and McKenzie showed that frame identities can be characterized by Mal’tsev conditions. Although that time there was a hope that their method combined with [17] gives a Mal’tsev condition for each λ that implies modularity, cf. [11, p. 155], Pálfy and Szabó [21] destroyed this expectation. The goal of the present paper is to prove that each lattice identity implying modularity is equivalent to a Mal’tsev condition. Moreover, this Mal’tsev condition is very easy to construct. In order to formulate a slightly stronger statement, some definitions come first. A lattice identity λ is said to imply modularity in congruence varieties, in notation λ |=c mod, if for any variety V if all the congruence lattices Con(A), A ∈ V , satisfy λ then all these lattices are modular. If λ implies modularity in the usual lattice theoretic sense then of course λ |=c mod as well. However, it was a great surprise by Nation [20] that λ |=c mod is possible even when λ does not imply modularity in the usual sense. Jónsson [13] gives an overview of similar results. We mention that there is an algorithm to test if λ |=c mod, cf. [5], which is based on Day and Freese [10]. Given a lattice term p and k ≥ 2, we define p via induction as follows. If p is a variable then let p = p. If p = r ∧ s then let p = r ∩ s. Finally, if p = r ∨ s then let p = r ◦s ◦r ◦s ◦ . . . with k factors on the right. When congruences or, more generally, reflexive compatible relations are substituted for the variables of p then the operations ∩ and ◦ will be interpreted as intersection and relational product, respectively. Now and in the sequel by a lattice identity λ we mean an inequality p ≤ q where p and q are lattice terms. This does not hurt generality, for each p ≤ q is equivalent to an appropriate identity r = s modulo lattice theory and vice versa. If λ : p ≤ q is a lattice identity and m,n ≥ 2 then we can consider the inclusion p ⊆ q. If A is an algebra then p and q do not give congruences in general when their variables are substituted by congruences of A. However, it makes sense to say that p ⊆ q holds or fails for congruences of A. Now Wille [24] and Pixley [22] give an easy algorithm to construct a strong Mal’tsev condition M(p ⊆ q) such that, for any variety V, p ⊆ q holds for congruences of all algebras in V if and only if M(p ⊆ q) holds in V. (Notice that the construction of M(p ⊆ q) is outlined in Freese and McKenzie [11, Chapter XIII], and, with the notation U(Gm(p) ≤ Gn(q)), it is detailed in [4].) Wille and Pixley showed also that p ⊆ q holds for congruences of algebras in V if and only if V satisfies the Mal’tsev condition ”there is an n such that M(p ⊆ q) holds”; this will be needed in our proof. Now we can formulate the main result. Theorem 1. Let λ : p ≤ q be a lattice identity such that λ |=c modularity. Then for any variety V the following two conditions are equivalent. (a) For all A ∈ V, λ holds in the congruence lattice of A. (b) V satisfies the Mal’tsev condition ”there is an n ≥ 2 such that M(p ⊆ q) and (Dn) hold”. This paper will not detail the construction of M(p ⊆ q), but we mention that if we consider λ : (x ∧ (y ∨ (x ∧ z)) ≤ (x ∧ y) ∨ (x ∧ z), the modular law, then Day’s characterization of congruence modularity becomes a particular case of Theorem 1. Before proving Theorem 1we give some definitions and remarks. Reflexive symmetric compatible relations of an algebra are called tolerances, cf. Chajda [1] for an overview. The set of tolerances of A will be denoted by TolA. The transitive closure of a tolerance Φ ∈ TolA will be denoted by