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

Abstract

For a partial order $⪯$ 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 $⪯$ 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.

arXiv comment7 pages