We show that non-trivial "way below" and "way above" relations on the posets of all pseudometrics and of all pseudoultrametrics on a fixed set $X$ are possible if and only if the set $X$ is finite.
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.
In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice.
A method of construction of an inverse system of finite compacta, indexed with positive reals, from a compact ultrapseudometric, is proposed. It is proved that homeomorphisms and natural preorder on the class of compact ultrapseudometrics correspond to strict isomorphisms and strict morphisms of the respective inverse systems. Properties of the poset obtained with "gluing together" of the isomorphic inverse systems, are discussed.
Approximation relations (``way below'' and ``way above'') onpointwise ordered sets of ultrapseudometrics are studied. It isproved that, to obtain continuity and/or dual continuity of theseposets in the sense considered in domain theory, i.e., possibilityto ``safely'' approximate all elements from below and from aboverespectively, one should restrict to compact ultrapseudometrics,not exceeding a fixed one. Such poset is a complete lattice,the partial order on it determines the Lawson topology and the dualLawson topology, which coincide (which is linked bicontinuity), arecompact Hausdorff, and agree with the uniform convergence metric.Necessary and sufficient conditions are proved for ``way below''and ``way above'' relations to hold.
MSC Classification: 06F30 , 54E35
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