Extension of Zorn's lemma to arbitrary binary relations

TL;DR

This paper extends Zorn's lemma to arbitrary binary relations, enabling its use in optimization and existence proofs even when transitivity is absent, broadening its applicability.

Contribution

It introduces a generalized version of Zorn's lemma applicable to non-transitive binary relations, facilitating new existence theorems.

Findings

Extended Zorn's lemma applicable to arbitrary binary relations

Proves existence of generalized solutions without maximal elements

Broadens the scope of Zorn's lemma in optimization

Abstract

In this note, the Zorn lemma is extended to arbitrary binary relations and thus the Zorn lemma can do for optimization when the transitivity is broken. Zorn's extended lemma can be used to prove existence theorems of generalized solution concepts for binary relations that do not have maximal elements.