Characterizing Tolerance Trivial Finite Algebras

An algebra A is tolerance trivial if Tol A = ConA where Tol A is the lattice of all tolerances on A. If A contains a Mal'cev function compatible with each T Tol A, then A is tolerance trivial. We investigate nite algebras satisfying also the converse statement. Let R be a binary relation on a set A and f be an n-ary function on A. We say that f is compatible with R or that R is compatible with f if Let A = (A; F) be an algebra. A reeexive and symmetric binary relation T on A compatible with each f 2 F is called a tolerance on A. The set of all tolerances on A is denoted by TolA. It is well-known that TolA is an algebraic lattice with respect to set inclusion. In general, the congruence lattice Con A is not a sublattice of TolA. If TolA = Con A, we say that an algebra A is tolerance trivial. A variety V is tolerance trivial if each A 2 V has this property. Tolerance trivial algebras were studied by numerous authors, see 7] for the rst essential results and 2] for the almost complete survey. Recent results on tolerance trivial BCK-algebras were published If a; b 2 A, denote by T(a; b) the least tolerance on the algebra A containing the pair ha; bi. Denote by _ the operation join in TolA; meet coincides with the set intersection. A function g, g : A n ! A, is called an n-ary algebraic function over A if there exists an (n + m)-ary term t over A (m 0) and elements a 1