Mal’cev conditions corresponding to identities for compatible reflexive relations

We investigate Mal’cev conditions described by those equations whose variables runs over the set of all compatible reflexive relations. Let p≤q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \le q$$\end{document} be an equation for the language {∧,∘,+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\wedge , \circ ,+\}$$\end{document}. We give a characterization of the class of all varieties which satisfy p≤q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \le q$$\end{document} over the set of all compatible reflexive relations. The aim is to find an analogon of the Pixley–Wille algorithm for conditions expressed by equations over the set of all compatible reflexive relations, and to characterize when an equation p≤q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \le q$$\end{document} expresses the same property when considered over the congruence lattices or over the sets of all compatible reflexive relations of algebras in a variety.


Introduction
Starting from Mal'cev's description of congruence permutability as in [16], the problem of characterizing properties of classes of varieties as Mal'cev conditions has led to several results. In [17] Pixley found a strong Mal'cev condition defining the class of varieties with distributive and permuting congruences. In [10] Jónsson shows a Mal'cev condition characterizing congruence distributivity, in [7] Day shows a Mal'cev condition characterizing the class of varieties with modular congruence lattices.
These results are examples of a more general theorem obtained independently by Pixley [18] and Wille [21] that can be considered as a foundational result in the field. They proved that if p ≤ q is a lattice identity, then the class of varieties whose congruence lattices satisfy p ≤ q is the intersection of countably many Mal'cev classes. [18] and [21] include an algorithm to generate Mal'cev conditions associated with congruence identities.
These researches have also led to the problem of studying equations where the variables run not only over the congruence lattices but in possibly strictly larger sets as the lattices of all tolerances or of all compatible reflexive relations. Results about this problem can be found in [6,8,9,12,14,15,20].
Let R and T be two binary relations on a set A. Then we define the k-fold relational product, in symbols • (k) , of R and T by: We denote by • the 2-fold relational product of two relations and we denote by The aim of this paper is to study the relationship between congruence equations and compatible reflexive relation equations. We denote by Crr(A) the set of all compatible reflexive relations of A. Let A be an algebra and let p and q be terms for the language {∧, +, •}, where ∧ is the set-theoretical intersection. Then we say that A satisfies the compatible reflexive relation equality (inequality) p ≈ q (p ≤ q) if and only if Crr(A) satisfies p ≈ q (p ≤ q). Let V be a variety. Then we say that V satisfies p ≈ q (p ≤ q) if and only if every algebra in V satisfies p ≈ q (p ≤ q). These equations are of interest since Mal'cev conditions are often express in term of this type of equations.
In Section 4 we study the connection between Mal'cev conditions and compatible reflexive relation inequalities. We provide a self-contained proof of an algorithm, claimed by Tschanz [19, pages 273-274] without a proof and similar to the one of Pixley-Wille [18], which allows us to compute a Mal'cev condition described by the property of satisfying a particular compatible reflexive relation equation. Theorem 1.1. Let p and q be two n-ary terms for the language {∧, •, + } with p +-free. Then the class of varieties such that p ≤ q holds in Crr(A) for all A ∈ V is a Mal'cev class. Furthermore, if q is +-free then the class of varieties such that p ≤ q holds in Crr(A) for all A ∈ V is a strong Mal'cev class.
Our contribute will be the proof of this theorem that provides several examples of descriptions of Mal'cev conditions. In Section 5 we give a characterization of an interesting set of equations which describe the same Mal'cev condition when considered with variables that run over the set of all compatible reflexive relations or over the congruence lattices.
In [13] it has been shown that, under a weak assumption on the term p, a variety satisfies the congruence identity p(α 1 , . . . , α n ) ≤ q(α 1 , . . . , α n ) if and only if it satisfies the tolerance identity p(T 1 , . . . , T n ) ≤ q(T 1 , . . . , T n ), provided we restrict the set of the variables to run over the representable tolerances, i.e. tolerances of the form R •R −1 where R is a compatible reflexive relation. Our main result, Theorem 5.1, provides a sufficient criterion to check whether an inequality p ≤ q describes the same Mal'cev condition considered as compatible reflexive relation inequality and as congruence inequality. The hypothesis of this theorem is quite restrictive for p and q but, in Section 5, we provide two examples showing that is quite difficult to extend this hypothesis.

Notation
Let A be an algebra and let X ⊆ A 2 . We denote by Crg A (X) the compatible reflexive relation generated by the set of pairs X. Let R and L be two binary relations on A. Furthermore, we denote by Eqv(R) the equivalence relation generated by a relation R.
Let p be a term for the language {∧, +, •}. Let k ∈ N. We denote by p (k) the {∧, •}-term obtained from p substituting any occurrence of + with the k-fold relational product • (k) . We

Labelled graphs and regular terms
In order to show the main results we recall the definition of a regular term introduced in [13]. Let p be a term for the language {•, ∧}. We define inductively the set L p and R p of variables on the left and right side of p respectively.
Definition 3.1. The class of regular terms is the smallest class of {•, ∧}-terms that: (1) contains all the variables; (2) contains p = q • r whenever q and r are regular and R q ∩ L r = ∅; (3) contains p = q ∧ r whenever q and r are regular, L q ∩ L r = ∅, and

Hence examples of regular terms are
are not regular. We can see that the majority of the known Mal'cev conditions given by congruence equations can be characterized by a congruence equation of regular terms.
The definition of a regular term can be better understood by means of the notion of the labelled graph associated with a term as in [2,3,4,5], and [12]. We show how to build the labelled graph associated with a {•, ∧}-term p.
Definition 3.2. Let S be a set of labels. Then a labelled graph is a directed graph (V, E) with a labelling function l : E → S.
We denote by (v 1 , v 2 ) an edge connecting the vertex v 1 to v 2 . Following [12], let p be a {∧, •}-term. We start with the graph G 1 (p) having an edge (y 1 , y 2 ), labelled with p, connecting two vertices y 1 and y 2 .
Starting from G 1 (p) we build a finite sequence of graphs G 1 (p), . . . , G l (p) = G(p) such that for all i ∈ [l] we select an edge (y j , y k ), connecting vertices y k and y j , from G i (p) labelled with a term w that is not a variable. Then we have two cases: by replacing the edge (y j , y k ) labelled w with two edges (y j , y k ) labelled u and v respectively, and both connecting the same vertices. If by introducing a new vertex y t and replacing the edge labelled w with two edges (y j , y t ) and (y t , y k ), labelled u and v respectively, and connecting the same vertices in serial through the new vertex. The sequence ends when all the edges are labelled with variables.
We can observe that a term is regular if in its graph, all the edges adjacent to any given vertex are labelled differently.
We denote by Y = {y 1 , . . . , y m } the set of vertices and we denote by capital letters the variables of p labelling the graph G(p). The main reason to introduce G(p) is stated in [5, Proposition 3.1] and in Claim 4.8 of [12]. The latter can be generalized to tolerances as in [13, Proposition 2.1] and also to relations in general. (1) Let Y → A : y s → a s be an assignment such that for all edges (y i , y j ) with label X k of G(p), we have (a i , a j ) ∈ R k . Then (a 1 , a 2 ) ∈ p(R 1 , . . . , R n ). (2) Conversely, given any (a 1 , a 2 ) ∈ p(R 1 , . . . , R n ), there is an assignment Y → A : y s → a s extending y 1 → a 1 , y 2 → a 2 such that (a i , a j ) ∈ R k whenever (y i , y j ) is an edge labelled with X k of G(p), where (y 1 , y 2 ) is the only edge of the graph G 1 (p).

The Algorithm
In this section we show the Algorithm in [18] and [21] and we then modify the last part of it in order to obtain a Mal'cev condition describing the class of varieties that satisfy the compatible reflexive relation equation p ≤ q. Let p ≤ q be an equation for the language {∧, •}. Let G(p) and G(q) be obtained from p and q (Definition in Section 2) with the procedure in Section 3. Then we define the sets of pairs: y j ) is an edge of G(p) with label X s }; where the elements {t 1 , . . . , t l } of the pairs in T t s (q) are m-ary terms with t 1 = π m 1 and t 2 = π m 2 in the variables {x 1 , . . . , x m }, m is the number of vertices in G(p), and where π m i is the m-ary term x i . Algorithm 4.1. Let p ≤ q be an equation for the language {∧, •}. Let G(p) and G(q) be obtained from p and q with the procedure in Section 3. Let us consider T s (p) and T t s (q) as in (4.1).
Next we define Eq(p ≤ q) as the set of all equations of the form: such that (t i , t j ) ∈ T t s (q) and the vector of indices (i 1 , . . . , i m ) ⊆ N n satisfies . This means that the variables that are in the equivalence relation generated by the pairs in T s (p) are collapsed.
We modify the last part of the Algorithm in [18,21] in order to obtain a set of equations Eq R (p ≤ q) that characterizes the Mal'cev condition describing the class of varieties which satisfy p ≤ q over the set of all compatible reflexive relations.

Algorithm 4.2. Let p ≤ q be an equation for the language {∧, •}. Let G(p)
and G(q) be obtained from p and q with the procedure in Section 3. Let us consider T s (p) and T t s (q) as in (4.1).
Let c(s) be the cardinality of T s (p) and let us set an order of these pairs {(y it , y jt )} 1≤t≤c(s) = T s (p). We define Eq R (p ≤ q) as the set of all equations of the form: . . . , x m , y j1 , . . . , y j c(s) ) ≈ t j (x 1 , . . . , x m ), such that (t i , t j ) ∈ T t s (q), t (i,j,s) is an (m + c(s))-ary term.
Note that the c(s)! ways to generate the equations given by (t i , t j ) ∈ T t s (q) and T s (p) in the Algorithm 4.2 give different terms t (i,j,s) s but equivalent conditions, up to reordering of the variables. Proof. The proof follows from the definition of regular term since in a graph of a regular term there cannot be two adjacent edges labelled with the same variable.
The next lemma is the main ingredient to prove Theorem 5.1. Indeed, this allows us to understand some properties fulfilled by Mal'cev conditions described by congruence inequalities with at most one occurrence of • in the right hand side of the equation.
Let a be a vector and let R be an irreflexive and antisymmetric relation over the set of all components of a which generates an equivalence relation with classes of at most 2 elements. This means that Eqv(R) is associated with a graph whose edges are disjoint and undirected. Let us denote by a (R,l) and a (R,r) the vectors such that if ((a) i , (a) j ) ∈ R, then (a (R,l) ) i = (a (R,l) ) j = (a) i , (a (R,r) ) i = (a (R,r) ) j = (a) j . If (a) i does not occur in pairs of R, then (a (R,l) ) i = (a (R,r) ) i = (a) i .

Lemma 4.4.
Let p, q be terms for the language {•, ∧}, with p regular. Let (t i , t j ) ∈ T t k (q) with k ∈ N. Then the following equations are logical consequences of Eq(p ≤ q): The next lemma is what allows us to characterize conditions described by compatible reflexive relation equations and is an equivalent statement for compatible reflexive relations of what is also called Mal'cev argument [1, Lemma 12.1]. With these tools we are ready to show an analogon of the Pixley-Wille algorithm [18,21]  Proposition 4.6. Let p and q be two n-ary terms for the language {∧, •, +} with p +-free. The class of varieties such that p ≤ q holds in Crr(A) for all A ∈ V is a Mal'cev class described by the family of equations {Eq R (p ≤ q (k) )} k∈N\{1} . Furthermore, if q is +-free is a strong Mal'cev class described by Eq R (p ≤ q).

Lemma 4.5. Let V be a variety of type F and let F V (X) be the free algebra generated by
Proof. Suppose that p ≤ q holds for the set of all the compatible reflexive relations of algebras in V, with p and q n-ary terms. Let T i (p) be constructed from p as in (4.1) and let V = {y 1 , . . . , y m } be the set of vertices of G(p), the graph associated with p. Let F = F V ({x 1 , . . . , x m }) be the m-generated free algebra in V. Then for all i = 1, . . . , n let R i be the compatible reflexive relation of F defined by: Then, from Proposition 3.3 item (1), it follows that: through the assignment y s → x s . Hence there exists a k ∈ N\{1} such that (x 1 , x 2 ) ∈ q (k) (R 1 , . . . , R m ). We prove that V satisfies Eq R (p ≤ q (k) ). To this end let Z = {z 1 , . . . , z u } be the set of vertices of G(q (k) ). Let us consider the pair of equations such that (t i , t j ) ∈ T t s (q (k) ), t (i,j,s) is an (m + c(s))-ary term, where c(s) = |T s (p)|, and the set of pairs {(y it , y jt )} 1≤t≤c(s) = T s (p). We know that (x 1 , x 2 ) = (π m 1 , π m 2 ) ∈ q (k) (R 1 , . . . , R n ) thus, by (2) of Proposition 3.3, there is an assignment z s → r s (x 1 , . . . , x m ) from Z → F that extends z 1 → x 1 , z 2 → x 2 such that (r i , r j ) ∈ R s whenever (z i , z j ) is an edge labelled with X s of G(q (k) ). Hence (r i , r j ) ∈ R s implies, by Lemma 4.5, that there exists an (m + c(s))- Conversely, suppose that there exists k ∈ N\{1} such that Eq R (p ≤ q (k) ) holds in V. Then let A ∈ V, a 1 , a 2 ∈ A, and let R 1 , . . . , R n ∈ Crg(A) be such that: (a 1 , a 2 ) ∈ p(R 1 , . . . , R n ) we want to prove that (a 1 , a 2 ) ∈ q (k) (R 1 , . . . , R n ).
From (2) of Proposition 3.3 we have that there exists an assignment ψ : Y → A extending y 1 → a 1 ,y 2 → a 2 such that y s → a s , with (a i , a j ) ∈ R s whenever (y i , y j ) is an X s -labelled edge of G(p).
Let Z = {z 1 , . . . , z u } be the set of vertices of G(q (k) ) and let |T s (p)| = c(s). From the hypothesis we have that for all (t i , t j ) ∈ T s (q (k) ), there exists an m + c(s)-ary term t (ti,tj ,s) such that:

Equations equivalent over congruences and over compatible reflexive relations
In this section we give a characterization of an interesting set of equations which describe the same Mal'cev condition when considered with variables that run over the set of all compatible reflexive relations or over the congruence lattices. In [13] Lipparini showed that if p is a regular term, a variety satisfies the congruence identity p(α 1 , . . . , α n ) ≤ q(α 1 , . . . , α n ) if and only if satisfies the tolerance identity p(T 1 , . . . , T n ) ≤ q(T 1 , . . . , T n ), provided we restrict the set of the variables to run over the representable tolerances, i.e. tolerances of the form R • R −1 where R is a compatible reflexive relation. It is also proved that both the properties to be regular for a term and to be representable for tolerances are not so restrictive. Instead, in the case of the compatible reflexive relations, the hypotheses on the equation p ≤ q are restrictive.