Sectionally Pseudocomplemented Posets

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.

Filters and congruences in sectionally pseudocomplemented lattices and posets

“…1: This lattice is not distributive and hence not relatively pseudocomplemented (see Birkhoff (1979)). The operation table for the sectional pseudocomplementation is as follows: Chajda et al (2021), Theorems 2.5 and 2.6, the following important result.…”

“…The following results were proved in Chajda et al (2021). Lemma 3.2 If P = (P, ≤, * ) is a sectionally pseudocomplemented poset with greatest element 1 and a, b, c ∈ P, then…”

“…It should be noted that there are sectionally pseudocomplemented posets which are not strongly sectionally pseudocomplemented, see, e.g., Chajda et al (2021), but these are rather curious.…”

“…This was realized by the first author in Chajda (2003) by introducing sectional pseudocomplementation. Later on, the concept of sectional pseudocomplementation was extended also to posets, see Chajda et al (2021).…”

“…For lattices we can use the machinery of universal algebra (see, e.g., Chajda et al (2012)) because sectionally pseudocomplemented lattices form a variety which is congruence permutable, congruence distributive and weakly regular. The situation with sectionally pseudocomplemented posets is a bit more complicated due to the fact that such a poset in general cannot be extended to a sectionally pseudocomplemented lattice by means of the Dedekind-MacNeille completion, see Chajda et al (2021) for the result.…”