I am dealing with not necessarily transitive relations satisfying, however, a well-ordering-like property. i.e., for any subset there exist a maximum element.
In mathematical terms, say $R \subset X \times X$ is my relation, and for any $Y \subset X$ the set of maximal elements $max_R(Y) \equiv \{ y \in Y | y R z, \forall z \in Y \} $ is non-empty.
Is there any name of this property of relation R? It seems to be strange to call it well-ordering property, when the relation itself is not ordering (terminology for ordering or lattices is useless since no transitivity is assumed).
Thank you.