Partitions of a Finite Partially Ordered Set

TL;DR

This paper explores three types of partitions of finite posets—monotone, regular, and open—providing multiple equivalent definitions and characterizations for each, enhancing understanding of their structure and properties.

Contribution

It introduces and proves the equivalence of three different notions of poset partitions, offering new combinatorial and relational characterizations.

Findings

Three notions of poset partitions are defined and shown to be equivalent.

Each partition type has multiple equivalent characterizations.

The work provides a comprehensive framework for understanding poset partitions.

Abstract

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some surjection having the poset as domain. We then obtain combinatorial characterisations of such notions in terms of blocks, without reference to surjection. Finally, we give a further, equivalent definition of each kind of partition by means of analogues of equivalence relations.