Congruence permutability for algebras with pseudocomplementation

In this note we are concerned with the permutability of congruence relations on semilattices and lattices with pseudocomplementation. There are some results in the literature along these lines. For example, in (8) H. P. Sankappanavar characterises those pseudocomplemented semilattices whose congruence lattice is modular and employs the result in conjunction with the well-known fact that algebras with permuting congruences are congruence-modular to characterise those pseudocomplemented semilattices with permuting congruences. Our first result is a direct, short proof of his result. In (2), J. Berman shows that for all congruences on a distributive lattice L with pseudocomplementation to permute it is necessary and sufficient that D(L), the dense filter of L, be relatively complemented. Our second result is a generalisation of that result to an important equational class of lattices with pseudocomplementation which properly contains the modular lattices with pseudocomplementation.