Closure operators in the category of quandles

TL;DR

This paper investigates a specific closure operator in the category of quandles, revealing its properties, relation to trivial quandles, and implications for algebraically connected and separated quandles.

Contribution

It provides a detailed analysis of the regular closure operator in quandles, showing its equivalence with the pullback closure and characterizing separated objects.

Findings

Closure operator coincides with the pullback closure operator.

Algebraically connected quandles form a connectedness related to trivial quandles.

Separated objects for the closure operator are exactly trivial quandles.

Abstract

We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analyze some of its properties. The category of algebraically connected quandles turns out to be a connectedness in the sense of Arhangel'ski\v{\i} and Wiegandt corresponding to the full subcategory of trivial quandles, while the disconnectedness associated with it is shown to contain all quasi-trivial quandles. The separated objects for the pullback closure operator are precisely the trivial quandles. A simple formula describing the effective closure operator on congruences corresponding to the same reflector is also given.