Extension of sectional pseudocomplementation in posets

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