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

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

2011-11-06

arxiv.org/abs/1111.1390v1

Mathematics

Functional Analysis

7 pages

Scientific paper

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.

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