Extension of sectional pseudocomplementation in posets

Abstract

Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation $\ast$ which associates with every pair $(x,y)$ of elements, where $x\ge y$, the pseudocomplement $x\ast y$ of $x$ in the upper section $[y)$. Any total extension $\to$ of $\ast$ 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.

arXiv commentpdfLaTeX, 28 pages, contains LaTeX figures and tables. V2: Proposition 3.6 corrected, the final part of Section 3.3 reorganized, Remark 3 edited, Theorem 7.6 and Corollary 7.7 strengthened, proof of Theorem 7.10 edited