A Note on Congruence Systems of Ms-algebras

Let A be an algebra. We use Con(A) to denote the congruence lattice of A. We say that θ, δ ∈ Con(A) permute if θ∨δ = {(x, y) ∈ A: there is z ∈ A such that (x, z) ∈ θ and (z, y) ∈ δ}. The algebra A is congruence permutable (permutable for short) if every pair of congruences in Con(A) permutes. By a system on A we understand a 2n-tuple (θ1, . . . , θn;x1, . . . , xn), where θ1, . . . , θn ∈ Con(A), x1, . . . , xn ∈ A and (xi, xj) ∈ θi∨θj for every 1 6 i, j 6 n. A solution of a system (θ1, . . . , θn;x1, . . . , xn) is an element x ∈ A such that (x, xi) ∈ θi for every i = 1, . . . , n. We note that if A is congruence permutable and Con(A) is distributive, then every system on A has a solution (folklore). An algebra 〈L,∧,∨, , 0, 1〉 of type (2, 2, 1, 0, 0) is an MS-algebra if it satisfies the following conditions: 〈L,∧,∨, 0, 1〉 is a bounded distributive lattice