Abstract
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 N5. 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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The authors would like to express their gratitude to the anonymous reviewer whose thorough and detailed comments led to a considerable improvement of this paper.
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Support of the research of the first two authors by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L, entitled “The many facets of orthomodularity”, as well as by ÖAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion”, and, concerning the first author, by IGA, project PřF 2020 014, is gratefully acknowledged. The third author was supported by the project “New approaches to aggregation operators in analysis and processing of data”, Nr. 18-06915S, by Czech Grant Agency (GAČR).
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Chajda, I., Länger, H. & Paseka, J. Sectionally Pseudocomplemented Posets. Order 38, 527–546 (2021). https://doi.org/10.1007/s11083-021-09555-6
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DOI: https://doi.org/10.1007/s11083-021-09555-6