On extension of partial orders to total preorders with prescribed symmetric part

TL;DR

This paper establishes a necessary and sufficient condition for extending a partial order to a total preorder with a prescribed symmetric part, generalizing Szpilrajn's theorem to include symmetric relations.

Contribution

It introduces a new criterion for extending partial orders to total preorders with specific symmetric parts, broadening classical order extension results.

Findings

Derived a necessary and sufficient condition for such extensions.

Generalized Szpilrajn theorem to include symmetric parts.

Provided a framework for extending partial orders with prescribed symmetric relations.

Abstract

For a partial order $\preceq$ on a set X and an equivalency relation S defined on the same set X we derive a necessary and sufficient condition for the existence of such a total preorder on X whose asymmetric part contains the asymmetric part of the given partial order $\preceq$ and whose symmetric part coincides with the given equivalence relation S. This result generalizes the classical Szpilrajn theorem on extension of a partial order to a perfect (linear) order.