Sectionally Pseudocomplemented Posets

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Publikace nespadá pod Fakultu sportovních studií, ale pod Přírodovědeckou fakultu. Oficiální stránka publikace je na webu muni.cz.
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CHAJDA Ivan LÄNGER Helmut PASEKA Jan

Rok publikování 2021
Druh Článek v odborném periodiku
Časopis / Zdroj Order
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www https://link.springer.com/article/10.1007/s11083-021-09555-6
Doi http://dx.doi.org/10.1007/s11083-021-09555-6
Klíčová slova Sectional pseudocomplementation; Poset; Congruence; Dedekind-MacNeille completion; Generalized ordinal sum
Popis The concept of a sectionally pseudocomplemented lattice was introduced in Birkhoff (1979) as an extension of relative pseudocomplementation for not necessarily distributive lattices. The typical example of such a lattice is the non-modular lattice N-5. The aim of this paper is to extend the concept of sectional pseudocomplementation from lattices to posets. At first we show that the class of sectionally pseudocomplemented lattices forms a variety of lattices which can be described by two simple identities. This variety has nice congruence properties. We summarize properties of sectionally pseudocomplemented posets and show differences to relative pseudocomplementation. We prove that every sectionally pseudocomplemented poset is completely L-semidistributive. We introduce the concept of congruence on these posets and show when the quotient structure becomes a poset again. Finally, we study the Dedekind-MacNeille completion of sectionally pseudocomplemented posets. We show that contrary to the case of relatively pseudocomplemented posets, this completion need not be sectionally pseudocomplemented but we present the construction of a so-called generalized ordinal sum which enables us to construct the Dedekind-MacNeille completion provided the completions of the summands are known.
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