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arXiv · 2512.17749

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

Gi-Sang Cheon·Samuele Giraudo·Gukwon Kwon·Hojoon Lee
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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.

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Gi-Sang Cheon, Samuele Giraudo, Gukwon Kwon, Hojoon Lee. 2025-12-19. Generating naturally labeled posets through matrix extensions, order ideals and automorphism groups. https://arxiv.org/abs/2512.17749

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