Topologies, Posets and Finite Quandles

TL;DR

This paper explores the relationship between topologies, posets, and finite quandles, establishing conditions for continuous functions and classifying topologies on finite quandles, with explicit computations for small cases.

Contribution

It characterizes when finite quandles admit non-trivial topologies with continuous right multiplications and classifies such topologies for specific classes like dihedral quandles.

Findings

Non-trivial topology exists on finite quandles with more than one orbit.

Right continuous posets on n-orbit quandles are n-partite.

Explicit counts of topologies for even dihedral 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 $T_0$-spaces and partially ordered sets (posets). We investigate Alexandroff $T_0$-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 given.