Varieties with modular and distributive lattices of symmetric or reflexive relations
Ivan Chajda · Czechoslovak Mathematical Journal · 1992
A binary relation R on an algebra (A, F) is called compatible if R satisfies the Substitution Property with respect to F, i.e. if for each n-ary / £ F, (a,-, 6,) £ R for i = 1, ..., n imply (f(a\,..., a n ), /(&i,..., 6 n )) G R. It was shown in [1] that for any subcollection C of the properties: reflexivity, symmetry, transitivity, the set of all compatible relations on (A, F) satisfying C forms an algebraic lattice (with respect to set inclusion).The modularity or distributivity of such lattices were characterized by some authors, especially for varieties of algebras.For congruences (i.e.reflexive, symmetric and transitive compatible relations), it was done by A. Day [5] and B. Jonsson [6], For tolerances (i.e.reflexive and symmetric compatible relations), it was solved in [2], For quasiorders (i.e.reflexive and transitive compatible relations), the distributivity was characterized in [4].For weak congruences (symmetric and transitive compatible relations), the answer has been given recently by G. Vojvodic and B. Seselja in [8].For general compatible relations, the solution is contained in [3], The aim of this paper is to characterize varieties whose members have distributive or modular lattices of symmetric or reflexive compatible relations.Notation.An algebra and its support will be denoted by the same letter.Let A be an algebra.Denote by Sym(.rl) the lattice of all symmetric compatible relations on A. Clearly, the empty relation is the least and A 2 is the greatest element of Sym(v4).The operation A (meet) in Sym(^4) coincides with set intersection.Denote by V the join in Sym(.A).For a, b E A denote by 5(a, 6) the least element of SyrmM) containing the pair (a, b).If £i, ..., x n are elements of A y denote by x the sequence X\y . .., Xn.