On the Extension of Quasipreferences to Weak Preferences with a Prescribed Indifference Relation

The topic of this paper is related to the Szpilrajn theorem and its generalizations on the extension of a binary relation to a perfect (linear) order or a total preorder. The main result is a necessary and sufficient condition for the existence of an extension of an asymmetric binary relation, called here a quasipreference, to a weak preference (an asymmetric and negatively transitive binary relation) whose indifference relation is equal to a beforehand given equivalence relation.