A Mathematica package to cope with partially ordered sets

TL;DR

This paper introduces a specialized Mathematica package designed for advanced manipulation and analysis of partially ordered sets, including features for enumeration, lattice construction, and categorical operations, tailored for research needs.

Contribution

The paper presents a new Mathematica package with tools specifically developed for studying partially ordered sets, filling gaps in existing combinatorial software.

Findings

Includes functions for basic PO set operations

Enables enumeration and visualization of partitions

Supports computation of products and coproducts in categories

Abstract

Mathematica offers, by way of the package Combinatorics, many useful functions to work on graphs and ordered structures, but none of these functions was specific enough to meet the needs of our research group. Moreover, the existing functions are not always helpful when one has to work on new concepts. In this paper we present a package of features developed in Mathematica which we consider particularly useful for the study of certain categories of partially ordered sets. Among the features offered, the package includes: (1) some basic features to treat partially ordered sets; (2) the ability to enumerate, create, and display monotone and regular partitions of partially ordered sets; (3) the capability of constructing the lattices of partitions of a poset, and of doing some useful computations on these structures; (4) the possibility of computing products and coproducts in the category of partially ordered sets and monotone maps; (5) the possibility of computing products and coproducts in the category of forests (disjoint union of trees) and open maps (cf. [DM06] for the product between forests).