Permutability of Tolerances with Factor and Decomposing Congruences

A variety V has tolerances permutable with factor congruences if for any A1, A2 of V and every tolerance T on A1 × A2 we have T ◦ Π1 = Π1 ◦ T and T ◦ Π2 = Π2 ◦ T , where Π1,Π2 are factor congruences. If B is a subalgebra of A1×A2, the congruences Θi = Πi∩B are called decomposing congruences. V has tolerances permutable with decomposing congruences if T ◦ Θi = Θi ◦ T (i = 1, 2) for each A1, A2 ∈ V , every subalgebra B of A1×A2 and any tolerance T on B. The paper contains Mal’cev type condition characterizing these varieties. Permutability of congruences with factor congruences was introduced by J. Hageman [5]: If A1, A2 are algebras of the same type, denote by Π1,Π2 the so called factor congruences on A1 × A2, i.e. Πi is a congruence induced by the i-th projection of A1 × A2 onto Ai (i = 1, 2). A variety V has congruences permutable with factor congruences if for any A1, A2 of V and each Θ ∈ ConA1 ×A2, Θ ◦Π1 = Π1 ◦Θ and Θ ◦Π2 = Π2 ◦Θ . The paper [4] contains a Mal’cev type characterization of varieties satisfying this condition, see also [4] for some details. This property was studied by J. Duda [3] for 3-permutability. In this paper we generalize the original property for tolerances and we give Mal’cev conditions characterizing these varieties. By a tolerance on an algebra (A,F ) is meant a reflexive and symmetric binary relation on A satisfying the substitution property with respect to all operations of F . The set of all tolerances on A forms a complete (even algebraic) lattice TolA with respect to set inclusion. Clearly, every congruence is a tolerance on A. The least element of TolA is the identity relation ωA, the greatest one is ιA = A×A. Hence, for every two elements a, b of A there exists the least tolerance on A containing the pair 〈a, b〉; it will be denoted by T (a, b) or T (〈a, b〉) and called the principal tolerance generated by 〈a, b〉. Received April 12, 1995. 1980 Mathematics Subject Classification (1991 Revision). Primary 08B05, 08A30.