# Sectionally pseudocomplemented posets

## TL;DR

This paper extends the concept of sectional pseudocomplementation from lattices to posets, explores their properties, congruences, and completions, and introduces generalized ordinal sums to construct Dedekind-MacNeille completions.

## Contribution

It generalizes sectional pseudocomplementation to posets and studies their algebraic properties, congruences, and completion methods, including a new construction technique.

## Key findings

- Sectionally pseudocomplemented lattices form a variety with simple identities.
- Every such poset is completely L-semidistributive.
- Dedekind-MacNeille completion may not preserve sectional pseudocomplementation, but generalized ordinal sums can be used for construction.

## Abstract

The concept of a sectionally pseudocomplemented lattice was introduced by I. Chajda 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 pseudocompelemented 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.