Some Separation Axioms in the category Ord of Ordered sets: Application in Alexandroff Topology, Specifically on Primal Spaces

Abstract

An order is any binary relation which is reflexive, transitive and anti-symmetric. Our main goal in this paper is to introduce some new binary relations between partial order and equality. Some implications between them and counterexamples ar clarified. Taking an Alexandroff $T_0$-space $(X,\tau)$, we study the order $\leq_{\tau}$. Finally, the particular case  $\leq_{f}$, where $f$ is a giving map from a set $X$ to itself, is explained.