Generating naturally labeled posets through matrix extensions, order ideals and automorphism groups

TL;DR

This paper introduces a matrix-based algebraic method for generating and enumerating naturally labeled posets, leveraging order ideals, automorphism groups, and topological lattice growth.

Contribution

It develops a systematic matrix extension approach, a sieve algorithm for poset generation, and a novel enumeration scheme based on distributive lattice growth.

Findings

Efficient generation of all admissible poset extensions using a sieve algorithm.

Characterization of valid poset matrices via fixed-point equations and order ideals.

A new constructive enumeration method based on the topological growth of distributive lattices.

Abstract

We propose a matrix approach for generating naturally labeled posets by representing each poset $P$ on the set $[n]$ as a Boolean poset matrix $A$. This algebraic representation enables a systematic handling of partial orderings through matrix extensions $A^v$. We show that $A^v$ defines a valid poset matrix if and only if the Boolean vector $v\in\mathbb B^n$ represents an order ideal of the poset $P$ associated to $A$, equivalently satisfying the fixed-point equation $vA=v$. Based on this characterization, we develop a sieve algorithm that generates all admissible extension vectors efficiently. Furthermore, we explore the twin-class decomposition of $A$, which partitions the elements of $P$ according to identical down- and up-sets. This structure provides an algebraic foundation for Burnside-type enumeration for Birkhoff's question on counting nonisomorphic posets on $[n]$ through the automorphism group ${\rm Aut}(A)$. Finally, we present an algorithmic generation scheme for the posets based on the topological growth of their distributive lattices, offering a new approach to constructive enumeration of poset families.