Topologies, posets and finite quandles

Abstract

An Alexandroff space is a topological space in which every intersection of open sets is open. There is one to one correspondence between Alexandroff Ty-spaces and partially ordered sets (posets). We investigate Alexandroff Ty-topologies on finite quandles. We prove that there is a non-trivial topology on a finite quandle making right multiplications continuous functions if and only if the quandle has more than one orbit. Furthermore, we show that right continuous posets on quandles with n orbits are n-partite. We also find, for the even dihedral quandles, the number of all possible topologies making the right multiplications continuous. Some explicit computations for quandles of cardinality up to five are givene prove that there is a non-trivial topology on a nite quandle making right multiplications continuous functions if and only if the quandle has more than one orbit. Furthermore, we show that right continuous posets on quandles with n orbits are n-partite. We also nd, for the even dihedral quandles, the number of all possible topologies making the right multiplications continuous. Some explicit computations for quandles of cardinality up to ve are given.