On congruence n-distributivity of ordered algebras (Q794684)

Nation proved in 1974 that n-distributive congruence varieties are distributive. The aim of the present paper is to generalize this result for the case of ordered algebras where an ordered algebra is a universal algebra which is partially ordered in such a manner that its operations are monotone with respect to the ordering. The main theorem of the paper asserts that if n is any positive integer then for classes which are closed with respect to subalgebras and direct products the following three conditions are equivalent: (i) the congruence lattices of all algebras in the class are n- distributive, (ii) the congruence lattices of all algebras in the class are distributive, (iii) there exists a sequence of ternary terms satisfying a certain system of equations. This result implies some known results on congruence distributive (n- distributive) varieties of universal algebras. The last part of the paper contains an algorithm which associates a strong Mal'cev type condition with an arbitrary lattice identity \(p\leq q\) such that the validity of the latter identity in some order-congruence lattices is equivalent to the validity of the former Mal'cev type condition in the corresponding variety of ordered algebras.