G. Czédli’s tolerance factor lattice construction and weak ordered relations

G. Czédli proved that the blocks of any compatible tolerance T of a lattice L can be ordered in such a way that they form a lattice L/T called the factor lattice of L modulo T. Here we show that the Dedekind–MacNeille completion of the lattice L/T is isomorphic to the concept lattice of the context (L, L, R), where R stands for the reflexive weak ordered relation ≤∘T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathord {\le } \circ T$$\end{document}. Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations R⊆L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\subseteq L^{2}$$\end{document} satisfying R=≤∘R∘≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\ \mathord {\le } \circ R\circ \mathord {\le } $$\end{document}.


Introduction
A binary relation on a (complete) lattice L is called (completely) compatible if it is a (complete) sublattice of the direct product L 2 = L × L. A reflexive symmetric and (completely) compatible relation T ⊆ L 2 is a called a (complete) tolerance of L. All tolerances of a lattice L, denoted by Tol(L) form an algebraic lattice (with respect to the inclusion).
Let T ∈ Tol(L) and X ⊆ L, X = ∅. If X 2 ⊆ T , then the set X is called a preblock of T . Blocks are maximal preblocks (with respect to ⊆). It is known that the blocks of any tolerance T are convex sublattices of L. In [1] G. Czédli proved that the blocks of T can be ordered in such a way that they form a lattice. This lattice is denoted by L/T and it is called the factor lattice of L modulo T . The notion of factor lattice constructed with his method is a natural generalization of factor lattice by a congruence. Definition 1.1. We say that a binary relation R is a weak ordered relation on the lattice L if it satisfies the following conditions: (1) for any u, x, y, z ∈ L, u ≤ x, (x, y) ∈ R and y ≤ z imply (u, z) ∈ R; (2) given any t ∈ L and any nonempty finite A ⊆ L, if (a, t) ∈ R holds for each a ∈ A then ( A, t) ∈ R; (3) given any z ∈ L and any nonempty finite A ⊆ L, if (z, a) ∈ R holds for each a ∈ A then (z, A) ∈ R.
Clearly, if conditions (2) and (3) hold for any two-element subset of L, then they hold for arbitrary finite and nonempty A ⊆ L.
An ordered relation R on a complete lattice L is a weak ordered relation which satisfies conditions (2) and (3) for arbitrary (i.e. even infinite or empty) A ⊆ L. This notion was introduced by S. Valentini [8], and in [5] it was shown that any ordered relation is a completely compatible relation on L. We will prove that reflexive weak ordered relations of L can be characterized as compatible reflexive relations R ⊆ L 2 satisfying R = ≤ • R • ≤. Moreover, we will see that for any T ∈ Tol(L), ≤ • T = ≤ • T • ≤ and R := ≤ • T is a weak ordered relation with the property that T = R ∩ R −1 , where R −1 stands for the inverse relation of R. The set of weak ordered relations and that of reflexive weak ordered relations of a lattice L will be denoted by WOR(L) and ReWOR(L), respectively.
The main results of the paper point out the connection between weak ordered relations and factor lattices defined by tolerances. It is proved that for any tolerance T of a lattice L the Dedekind-MacNeille completion of L/T is isomorphic to the concept lattice L(L, L, R) of the context (L, L, R), where R := ≤ • T . It is also shown that the blocks of T correspond exactly to the concepts (A, B) ∈ L(L, L, R) having the property that A ∩ B = ∅. This result generalizes a result of [6], where for any complete lattice L and any complete tolerance T ⊆ L 2 the isomorphism L/T ∼ = L(L, L, ≤ • T ) was established.
The paper is structured as follows: In Section 2 some basic notions and the interrelation between the lattices Tol(L) and ReWOR(L) are presented. In Section 3, the concept lattice L(L, L, ≤ • T ) is described and the main results of the paper are presented.
The relations ≤ and = L×L are examples of reflexive ordered relations. We consider the empty relation ∅ also as a weak ordered relation. Clearly, Vol. 82 (2021) Factor lattice construction Page 3 of 9 21 relation ∅ is not reflexive. Another weak ordered relation which is not reflexive in general, is given in the following Example 2.1. Let f : L → L be a join-endomorphism of the lattice L. Then the binary relation is a weak ordered relation on the lattice L.
We note that the above relation appears already in Valentini's paper [8] who proved that in case of a complete join-endomorphism R f is an ordered relation. Similarly, the "complete versions" of the claims listed in the next proposition appear in [8] with almost the same proofs.
Proof. Since The exprssion Wor(L) has to be formatted everywhere in the paper as in Proposition 2.2 Wor(L) is closed with respect to arbitrary intersections, including the empty intersection, it follows that (WOR(L), ⊆) is a complete lattice. Let R, S ∈ WOR(L). We prove that R • S ∈ WOR(L). Indeed, condition (1) of Definition 1.1 holds trivially. Conditions (2) and (3) are shown by using the same arguments as in the proof of the second lemma in [8], i.e. by using the compatibility of R and the fact that S satisfies condition (2), respectively (3). Now condition (1) yields ≤ • R ⊆ R and R • ≤ ⊆ R, for any R ∈ WOR(L). Since ≤ is a reflexive relation, the converse inclusions are clear. Thus we get (See the same claim and proof in [8, p. 2] for ordered relations.) Now take any R 1 , R 2 , S ∈ WOR(L), and prove identity (D1). The in- The proof of the first theorem in [8] uses the same argument to show that a stronger form of (D1) and (D2) holds for the ordered relations of a complete lattice. We note also that (D1) and (D2) are valid in any algebra with a (ternary) majority term. As a consequence of Proposition 2.2 we obtain 21 Page 4 of 9 S. Radeleczki Algebra Univers.

Corollary 2.3.
Let R be a binary relation on the lattice L. Then the following are equivalent: . We already know that any weak ordered relation R is compatible. As ≤ is the unit of the monoid (WOR(L),  The next result was established in [5, Thm. 12], for ordered relations. Since infinite joins and meets and the completeness of the lattice are not used in its proof, this remains valid also in our case, i.e, for reflexive weak ordered relations. We will show only that the maps below are well-defined. The following corollary is obvious:

(i) T = (≤ • T ) ∩ (T • ≥), for every T ∈ Tol(L).
(ii) Any reflexive weak ordered relation R ⊆ L 2 has the form R = ≤ • R ∩ R −1 , i.e. it can be derived from a tolerance S = R ∩ R −1 .   Proof. Suppose that x ≤ a for some a ∈ A and x ∈ L. Since (a, b) ∈ R for all a ∈ A and b ∈ B and R is a weak ordered relation, we obtain (x, b) ∈ R, for all b ∈ B. Hence x ∈ I B = A. Now let a 1 , a 2 ∈ A. Then for each b ∈ B the relations (a 1 , b) , (a 2 , b) ∈ R imply (a 1 ∨ a 2 , b) ∈ R (see Definition 1.1(2)). Hence a 1 ∨ a 2 ∈ I B = A . This proves that A is an ideal of L. The fact that B is a filter of L is proved dually.

Concept lattices induced by weak ordered relations
For any subset X ⊆ L of a lattice L, let [X) and (X] denote the filter and the ideal generated by X, respectively. We will use the following (see also [1,Lemma 3]).
and this means that C RR ∩ C R is a preblock of T . Since C is block and C ⊆ C RR ∩ C R , we obtain C = C RR ∩ C R . Because R is a weak ordered relation and (C RR , C R ) is a concept of the context (L, L, R), the extent C RR is an ideal of L and the intent C R is a filter of L, according to Proposition 3.1. As C is a convex sublattice of L, by using Lemma 3.
Then we get δ( . On the other hand, in view of [4] (or [9]), the join operation in L(L, L, R) has the form is proved dually. Thus δ is a lattice embedding.
We note that the embedding δ is not surjective in general. This is because the concept lattice L(L, L, ≤ • T ) is always complete, however L/T is not necessarily so, for example, if L is a non-complete lattice and T is its zero tolerance. Denote the Dedekind-MacNeille completion of a lattice L by DM(L). It is known that DM(L) is isomorphic to the concept lattice L(L, L, ≤), see e.g. [4]. Remark 3.7. This is the case when the factor lattice L/T is finite. The same result, i.e. L/T ∼ = L(L, L, ≤ • T ) we obtain also for a complete tolerance T of a complete lattice L, because then L/T is a complete lattice. This isomorphism for complete tolerances is also established in [6].