For a compact Hausdorff topological lattice L the set M L X of L-valued normed regular capacities on a compact Hausdorff topological space X is investigated. It is shown that the set M L X carries a compact Hausdorff topology, and M L extends to a weakly τ -normal functor in the category of compacta. If L is an upper and lower Lawson semilattice, then M L is the functorial part of two semimonads. These semimonads coincide and are a monad if and only if L is distributive, i.e., is a Lawson lattice. The obtained results have a natural interpretation if capacities are regarded as subjective estimates of likelihood of realization of events in conditions of uncertainty.
In this paper we study pseudoultrametrics, which are a natural mixture of ultrametrics and pseudometrics. They satisfy a stronger form of the triangle inequality than usual pseudometrics and naturally arise in problems of classification and recognition. The text focuses on the natural partial order on the set of all pseudoultrametrics on a fixed (not necessarily finite) set. In addition to the “way below” relation induced by a partial order, we introduce its version which we call “weakly way below”. It is shown that a pseudoultrametric should satisfy natural conditions closely related to compactness, for the set of all pseudoultrametric weakly way below it to be non-trivial (to consist not only of the zero pseudoultrametric). For non-triviality of the set of all pseudoultrametrics way below a given one, the latter must be compact. On the other hand, each compact pseudoultrametric is the least upper bound of the directed set of all pseudoultrametrics way below it, which are compact as well. Thus it is proved that the set CPsU(X) of all compact pseudoultrametric on a set X is a continuous poset. This shows that compactness is a crucial requirement for efficiency of approximation in methods of classification by means of ultrapseudometrics.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.