Infinite saturated orders

TL;DR

This paper extends the concept of saturated orders to infinite partial orders, providing set-theoretic and algebraic characterizations, and analyzes their proof-theoretic strength within reverse mathematics.

Contribution

It introduces a generalized notion of saturated orders for infinite cases and explores their equivalence and proof-theoretic strength in reverse mathematics.

Findings

Equivalence of characterizations varies with definitions

Proof strength is either in RCA_0 or ACA_0

Provides set-theoretic and algebraic frameworks for infinite saturated orders

Abstract

We generalize the notion of saturated order to infinite partial orders and give both a set-theoretic and an algebraic characterization of such orders. We then study the proof theoretic strength of the equivalence of these characterizations in the context of reverse mathematics, showing that depending on one's choice of definitions it is either provable in $\mathsf{RCA}_0$ or equivalent to $\mathsf{ACA}_0$.