Linear extensions of partial orders and Reverse Mathematics

TL;DR

This paper studies special classes of partial orders called au-like, exploring their linear extensions and embeddability, and establishes their logical strength within reverse mathematics frameworks.

Contribution

It introduces au-like partial orders and analyzes the reverse mathematical strength of related extension and embeddability statements.

Findings

Equivalent to B extSigma^0_2 over RCA_0

Equivalent to ACA_0 over RCA_0

New results for au = extzeta-like partial orders

Abstract

We introduce the notion of \tau-like partial order, where \tau is one of the linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example, being \omega-like means that every element has finitely many predecessors, while being \zeta-like means that every interval is finite. We consider statements of the form "any \tau-like partial order has a \tau-like linear extension" and "any \tau-like partial order is embeddable into \tau" (when \tau\ is \zeta\ this result appears to be new). Working in the framework of reverse mathematics, we show that these statements are equivalent either to B\Sigma^0_2 or to ACA_0 over the usual base system RCA_0.