Extension of sectional pseudocomplementation in posets

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.