Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and conco...

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. show that only countably many these can be virtually torsion-free. This gives new examples spaces are not torsion-free and cannot subgroups groups.

Sets with a self-distributive operation (in the sense of ( ◃ b ) c = ), in particular quandles, appear knot and braid theories, Hopf algebra classification, study Yang–Baxter equation, other areas. An important invariant quandles is their structure group. The group finite quandle known to be either “boring” (free abelian), or “interesting” (non-abelian torsion). In this paper we explicitly desc...