Germs in a poset

TL;DR

This paper introduces the concept of germs in finite posets and lattices, establishing the existence of a largest germ extension called germ closure, and characterizing germ extensible subsets within finite lattices.

Contribution

It defines germ and germ extension in finite posets, proves the existence of a maximal germ extension (germ closure), and characterizes germ extensible subsets in finite lattices.

Findings

Existence of a largest germ extension called germ closure.

Unique germ extensible subset for any subset of a finite lattice.

Characterization of germ extensible subsets via germ closure embedding.

Abstract

Motivated by the theory of correspondence functors, we introduce the notion of {\em germ} in a finite poset, and the notion of {\em germ extension} of a poset. We show that any finite poset admits a largest germ extension called its {\em germ closure}. We say that a subset $U$ of a finite lattice $T$ is {\em germ extensible} in $T$ if the germ closure of $U$ naturally embeds in $T$. We show that any for any subset $S$ of a finite lattice $T$, there is a unique germ extensible subset $U$ of $T$ such that $U\subseteq S\subseteq \overline{G}(U)$, where $\overline{G}(U)\subseteq T$ is the embedding of the germ closure of $U$.