ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 65,   1   (1996)
pp.   93-99

PERMUTABILITY OF TOLERANCES WITH FACTOR AND DECOMPOSING CONGRUENCES
I. CHAJDA

Abstract.  A variety $\vv$ has tolerances permutable with factor congruences if for any $A_1,A_2$ of $\vv$ and every tolerance $T$ on $A_1 \times A_2$ we have $T \circ \Pi_1 =\Pi_1 \circ T$ and $T \circ \Pi_2 =\Pi_2 \circ T$, where $\Pi_1,\Pi_2$ are factor congruences. If $B$ is a subalgebra of $A_1 \times A_2$, the congruences $\Theta_i= \Pi_i \cap B^2$ are called decomposing congruences. $\vv$ has tolerances permutable with decomposing congruences if $T \circ \Theta_i =\Theta_i \circ T$ $(i=1,2)$ for each $A_1,A_2 \in \vv$, every subalgebra $B$ of $A_1 \times A_2$ and any tolerance $T$ on $B$. The paper contains Mal'cev type condition characterizing these varieties.

AMS subject classification.  08B05, 08A30
Keywords.  Tolerance relation, direct product, factor congruence, permutability of congruences, Mal'cev condition