  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.
