Extension of sectional pseudocomplementation in posets

Abstract

Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation * which associates with every pair (x,y) of elements, where x ≥ y, the pseudocomplement x*y of x in the upper section [y). Any total extension → of * is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have already be studied on semilattices and lattices. We describe several naturally arising classes of general posets with extended sp-complementation, present respective elementary properties of this operation, demonstrate that two other known attempts to isolate particular such classes are in fact not quite correct, and suggest suitable improvements.MSC Classification: 03G25 , 06A99 , 08A99