A note on non-generators in partially ordered sets

TL;DR

This paper explores the concept of non-generators in arbitrary partially ordered sets, providing new characterizations and linking some results to fundamental set-theoretic axioms like Zorn's Lemma and the Axiom of Choice.

Contribution

It extends the characterization of non-generators from algebraic posets to general posets and discusses the set-theoretic implications of these characterizations.

Findings

Characterizations of non-generators in arbitrary posets

Some characterizations are equivalent to Zorn's Lemma

Working on closure spaces offers no essential advantage

Abstract

A folklore argument shows that Frattini's characterization of non-generators works in the framework of algebraic partially ordered sets. We provide characterizations of non-generators in arbitrary partially ordered sets. The validity of some characterizations is equivalent to Zorn's Lemma, hence to the Axiom of choice. We notice that working on closure spaces or posets provides no essential improvement.