Continuous Poset Research Articles

The Intrinsic Characterization of a Fuzzy Consistently Connected Domain

The concepts of a fuzzy connected set (fc set) and a fuzzy consistently connected set (fcc set) are introduced on fuzzy posets, along with a discussion of their basic properties. Inspired by some equivalent conditions of crisp connected sets, the characterizations of the fc sets are given, and we also explore fuzzy completeness and fuzzy compactness in addition to defining a new fuzzy way-below relation based on fcc complete sets. Using this relationship as a basis, the fcc domain is also provided and studied, and its equivalent characterizations are obtained. In summary, we develop a method to establish fcc completeness from a continuous poset.

Approximation Relations on the Posets of Pseudoultrametrics

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.

A note on “Posets having continuous interval”

In this note we present a counterexample for a result in [2], where the concept of continuous interval posets (CI-posets for short) was introduced. A CI-poset is a poset in which every nonempty interval [x,y] is a continuous poset (in the restricted order) and the above mentioned result is about continuous functions spaces [X⟶P], where X is a core compact topological space and P is a CI-poset (with its Scott topology).

SI-continuous spaces and continuous posets

Recently, Zhao and Ho introduced and studied SI-continuous spaces, which can be seen as topological counterparts of continuous posets. The main purpose of this paper is to investigate the relationships between SI-continuous spaces and continuous posets. We prove that a C-space is an SI-continuous space if and only if it is a continuous poset under the specialization order. Furthermore, we introduce the notion of strong SI-continuous spaces and construct an adjunction between the category of domains and the category of strong SI-continuous spaces.

Some results on fuzzy $$Z_{L}$$ Z L -continuous(algebraic) poset

In this paper, based on complete residuated lattices, the properties of fuzzy $$Z_{L}$$ -continuous posets and fuzzy $$Z_{L}$$ -algebraic posets are investigated. Then we show that the set of fuzzy $$Z_{L}$$ -ideals on the set of all compact elements of a fuzzy $$Z_{L}$$ -algebraic poset is a fuzzy $$Z_{L}$$ -closure system. Also, we prove that the image of a (strongly) fuzzy $$Z_{L}$$ -continuous poset under a fuzzy $$Z_{L}$$ -morphism is (strongly) fuzzy $$Z_{L}$$ -continuous.

Quasi-metrics and monotone normality

The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space.

Z-abstract Basis

In this paper, the notion of a Z-abstract basis is defined and its properties are discussed. It is proved that for a union-complete subset system, the basis of a Z-continuous poset, with the restricted ≪Z-relation, is a Z-abstract basis and the rounded Z-ideal completion of the Z-abstract basis is isomorphic to the Z-continuous poset. Conversely, the rounded Z-ideal completion of a Z-abstract basis is a Z-continuous poset.

Meet continuity properties of posets

In this paper we consider a number of properties of posets that are not directed complete: in particular, meet continuous posets, locally meet continuous posets and PI-meet continuous posets are introduced. Characterizations of (locally) meet continuous posets are presented. The main results are: (1) A poset is meet continuous iff its lattice of Scott closed subsets is a complete Heyting algebra; (2) A poset is a meet continuous poset with a lower hereditary Scott topology iff its upper topology is contained in its local Scott topology and the lattice of all local Scott closed sets is a complete Heyting algebra; and (3) A poset with a lower hereditary Scott topology is meet continuous iff it is locally meet continuous, iff it is PI-meet continuous.

Any [formula omitted] space has a continuous poset model

In this paper we prove that any T 1 subspace of a continuous dcpo with the relative Scott topology can be “modeled” by a continuous poset. Using this result we are able to show that any T 1 topological space ( X , τ ) is homeomorphic to the space of maximal elements of a continuous poset. We also find a bitopological characterization of a topological space ( X , τ ) that can be modeled by a continuous poset. It is proved that for any T 1 topological space ( X , τ ) there is a T 1 topology τ ∗ on X such that ( X , τ , τ ∗ ) is pairwise completely regular.

Completions of partial metrics into value lattices

In this paper we investigate some notions of completion of partial metric spaces, including the bicompletion, the Smyth completion, and a new “spherical completion”. Given an auxiliary relation, we show that it arises from a totally bounded partial metric space, and the spherical completion of such a space is its round ideal completion. We also give an example of a totally bounded partial metric space whose bicompletion and Smyth completion are not continuous posets. Finally, we present an example of a totally bounded partial metric giving rise to the Scott and lower topologies of a continuous poset, but whose spherical completion is not a continuous poset.

Generalized ultrametric spaces in quantitative domain theory

Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be defined for every metric space and that reflects many of its properties. On the other hand, in order to obtain a broader framework for applications and possible connections to domain theory, generalized ultrametric spaces ( gums) have been introduced. In this paper, we employ the space of formal balls as a tool for studying these more general metrics by using concepts and results from domain theory. It turns out that many properties of the metric can be characterized via its formal-ball space. Furthermore, we can state new results on the topology of gums as well as two new fixed point theorems, which may be compared to the Prieß -Crampe and Ribenboim theorem, and the Banach fixed point theorem, respectively. Deeper insights into the nature of formal-ball spaces are gained by applying methods from category theory. Our results suggest that, while being a useful tool for the study of gums, the space of formal balls does not provide the hoped-for general connection to domain theory.

Monoids over domains

In this paper, we describe three distinct monoids over domains, each with a commutative analog, which define bag domain monoids. Our results were inspired by work by Varacca (Varacca 2003), and they lead to a constructive approach to his Hoare indexed valuations over a continuous poset .

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are: (1) A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism; (2) A poset is continuous iff its Scott topology is completely distributive; (3) A topological T 0 space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology; (4) A topological T 1 space is a discrete space iff its topology is completely distributive. These results generalize the relevant results obtained by J.D. Lawson for dcpos.

Lim-inf convergence in partially ordered sets

The lim-inf convergence in a complete lattice was introduced by Scott to characterize continuous lattices. Here we introduce and study the lim-inf convergence in a partially ordered set. The main result is that for a poset P the lim-inf convergence is topological if and only if P is a continuous poset. A weaker form of lim-inf convergence in posets is also discussed.

Order Environments of Topological Spaces

In this paper it is proved that a space may be realized as the set of the maximal elements in a continuous poset if and only if it is Tychonoff.

Partial metrizability in value quantales

Partial metrics are metrics except that the distance from a point to itself need not be 0. These are useful in modelling partially defined information, which often appears in computer science. We generalize this notion to study “partial metrics” whose values lie in a value quantale which may be other than the reals. Then each topology arises from such a generalized metric, and for each continuous poset, there is such a generalized metric whose topology is the Scott topology, and whose dual topology is the lower topology. These are both corollaries to our result that a bitopological space is pairwise completely regular if and only if there is such a generalized metric whose topology is the first topology, and whose dual topology is the second.

Posets having continuous intervals

In this paper we consider posets in which each order interval [ a, b] is a continuous poset or continuous domain. After developing some basic theory for such posets, we derive our major result: if X is a core compact space and L is a poset equipped with the Scott topology (assumed to satisfy a mild extra condition) for which each interval is a continuous sup-semilattice, then the function space of continuous locally bounded functions from X into L has intervals that are continuous sup-semilattices. This substantially generalizes known results for continuous domains.

Extension of valuations on locally compact sober spaces

We show that every locally finite continuous valuation defined on the lattice of open sets of a regular or locally compact sober space extends uniquely to a Borel measure.In the sequel we derive a maximal point space representation for any locally compact sober space (X,G). That is, we show that there exists a continuous poset (∧ X, ⊆) such that X embeds as the subset of maximal elements of ΛX where the relative Lawson topology of ΛX induces the patch topology of X.We characterise the probabilistic power domain of a stably locally compact space as a stochastically ordered space of probability measures.

Measurements on domains and topology

The concept of a measurement on a continuous partial order has been recently introduced by Keye Martin. Measurement provides a uniform degree of approximation for elements of the kernal, i.e., those elements with measure zero. Measurement also induces a Scott topology for elements “near” the kernal. The results below are joint work with Keye Martin.IX is developable and T_1 iff it is the kernel of a measurement on a continuous poset, X is developable T1 and choquet complete iff it is the kernal of a measurement on a continuous cpo.IIFor each developable T1-space X, there exists a developable T1-space M(X) with a poset order < such that (1) X is the kernal of a measurement. (2) the topology on M(X) is exactly the topology induced by the measurement. (3) If X is T_2, then M(X) is a Moore space. (4) If X is a complete (or even semi-complete) Moore space then (M(X),<) is a cpo. (5) If X is the real line, then M(X) is a non-normal Moore space. (6) (MA + not CH) If X is a subspace of the real line and omega < card(X) < c, then M(X) is a normal nonmetrizable Moore space.IIIThe countable ordinals with the order topology is the top of a Scott domain, and is a G_delta set with respect to the Scott topology on the domain. [This answers several questions open questions in the area.]IVFinally, we give a new recursion induction theorem for cpo's using measurement theory.

The other topology

Understanding and in some cases, solution of problems involving topology in computer science, involve two interrelated topologies. A classical such case is that of the Scott and lower topologies on a continuous poset. A more recent example is the solution of the maximal point space problem (the maximal point spaces of Scott domains are the complete separable metrizable spaces). The role of these topologies will be discussed with reference to the two situations above and others.