Weak Relative Pseudocomplements in Semilattices

Weak relative pseudocomplementation on a meet semilattice S is a partial operation ∗ which associates with every pair (x, y) of elements, where x ≥ y, an element z (the weak pseudocomplement of x relative to y) which is the greatest among elements u such that y = u ∧ x. The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplement of x relative to y. A weakly relatively pseudomented semilattice is said to be extended, if it is equipped with a total binary operation extending ∗. We study congruence properties of the variety of such semilattices and review some of its subvarieties already described in the literature.