Extension of sectional pseudocomplementation in posets. (English) Zbl 07922995
Summary: 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 \(\rightarrow\) of \(*\) is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have already been 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:
| 06-XX | Order, lattices, ordered algebraic structures |
Keywords:
extended sectional pseudocomplementation; implicative poset; relative pseudocomplementation; sectional pseudocomplementation; weak relative pseudocomplementationReferences:
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