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Weak relative pseudocomplements in semilattices

Cirulis J.

Demonstratio Mathematica

2011

Vol. 44, nr 4

651-672

Weak relative pseudocomplementation on a meet semilattice S is a partial operation * which associates with every pair (x, y) of elements, where (…) an element z (the weak pseudocomplement of x relative to y which is the greatest among elements u such that (…). 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.

Subtraction-like operations in nearsemilattices

Cirulis J.

Demonstratio Mathematica

2010

Vol. 43, nr 4

725-738

A nearsemilattice is a poset having the upper-bound property. A binary operation — on a poset with the least element 0 is said to be subtraction-like if x ≤ y if and only if x — y = 0 for all x, y. Associated with such an operation is a family of partial operations lp defined by lp(x) := p— x on every initial segment [0, p]; these operations are thought of as local (sectional) complementations of some kind. We study several types of subtraction-like operations, show that each of these operations can be restored in a uniform way from the corresponding local complementations, and state some connections between properties of a (sufficiently strong) subtraction on a nearsemilattice and distributivity of the latter.