Trapezoid lemma and congruence distributivity
Ivan Chajda, Gábor Czédli, Eszter K. Horváth · Czech digital mathematics library · 2003
Motivated by Gumm's (rectangular) Shifting Lemma, in our con text a condition rather than a statement, and Shifting Principle, which play a key role in his treatment of congruence modularity and the theory of modular commutator, the present paper relates analogous triangular and trapezoid lem mas and principles to the distributivity of congruence lattices of single algebras and varieties.For varieties, the Trapezoid Lemma is equivalent to congruence distributivity.As a byproduct, congruence distributivity is characterized by a Mal'cev condition with a very clear connection with Day terms characterizing congruence modularity.Some results presented here were previously announced by J. Duda.H.-P. Gumm [11] defined a certain condition by a rectangular scheme for congruences, respectively congruences and tolerances of an algebra under the name Shifting Lemma and Shifting Principle.In a variety, each of these two conditions is equivalent to congruence modularity.Keeping congruence distribu tivity rather than congruence modularity in mind our goal is to study some other schemes which can be defined by triangles or trapezes.Following Gumm's style of [11; Corollary 3.6], schemes for congruences will be called lemmas although they are just conditions, and we keep the word principle for schemes where toler ances also occur.We are going to study our conditions for congruences of single algebras and for congruences of varieties.As it will be detailed at the end of the paper, some of our results have previously been announced by Duda [7].