Quasicontinuous Domains Research Articles

Upper powerdomains of quasicontinuous dcpos

The upper powerdomain of an arbitrary dcpo L can be constructed by the Scott completion of the directed upper powerspace of L. By means of this result, we investigate the upper powerdomains of quasicontinuous domains. The main results are: (1) The upper powerdomain PU(L) of a quasicontinuous domain L is quasicontinuous iff the directed upper powerspace of L is endowed with the Scott topology, iff PU(L) is isomorphic to Q(L), the semilattice of nonempty Scott compact saturated subsets of L. (2) There exists a quasicontinuous dcpo whose upper powerdomain is not quasicontinuous. (3) The upper powerdomain of a strongly quasicontinuous domain L is isomorphic to Q(L).

Quasiexact posets and the moderate meet-continuity

The study of weak domains and quasicontinuous domains leads to the consideration of two types generalizations of domains. In the current paper, we define the weak way-below relation between two nonempty subsets of a poset and quasiexact posets. We prove some connections among quasiexact posets, quasicontinuous domains and weak domains. Furthermore, we introduce the weak way-below finitely determined topology and study its links to Scott topology and the weak way-below topology first considered by Mushburn. It is also proved that a dcpo is a domain if it is quasiexact and moderately meet continuous with the weak way-below relation weakly increasing.

A COMPACT SPACE IS NOT ALWAYS SI-COMPACT

We consider the poset 𝒬(P) of all nonempty compact saturated subsets of the Scott space of a poset P, equipped with the reverse inclusion order. We also introduce the notion of T-lattices, that is, a complete lattice L is called a T-lattice if for any x∈L∖{1L}, ↑x∖{x}∈𝒬(L). We prove that if P and Q are quasicontinuous domains or T-lattices, P is order isomorphic to Q if and only if (𝒬(P),⊇) is order isomorphic to (𝒬(Q),⊇). We also prove that for a compact space X, if the Scott space of the open set lattice 𝒪(X) is sober, then X is SI-compact. Using this result and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not SI-compact. This gives a positive answer to an open problem posed by Zhao and Ho.

Coincidence of the Isbell and Scott topologies on the function spaces of quasicontinuous domains

The function space plays an indispensable role in investigating the Cartesian closedness of categories of domains. This paper focuses on the quasicontinuity and topological properties of the function spaces associated with quasicontinuous domains. For any quasicontinuous domain P with property M⁎ and bounded complete algebraic domain X, it is shown that: (i) the function space [X→P] is a quasicontinuous domain; especially, a counterexample is given to illustrate that property M⁎ is necessary for the quasicontinuity of the function space; (ii) the Isbell topology and Scott topology are equal in the function space [X→P].

Induced Topologies on the Poset of Finitely Generated Saturated Sets

In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset FinP of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any T0 space (X,τ) and U∈τ, we construct a topology τF generated by the basic open subsets UF={↑F∈FinX:F⊆U}. It is shown that a T0 space (X,τ) is a hypercontinuous lattice iff τF is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff FinP is hypercontinuous.

Categories of Locally Hypercompact Spaces and Quasicontinuous Posets

A subset of a topological space is hypercompact if its saturation (the intersection of its neighborhoods) is generated by a finite set. Locally hypercompact spaces are defined by the existence of hypercompact neighborhood bases at each point. We exhibit many useful properties of such spaces, often based on Rudin’s Lemma, which is equivalent to the Ultrafilter Principle and ensures that the Scott spaces of quasicontinuous domains are exactly the locally hypercompact sober spaces. We characterize their patch spaces (the Lawson spaces) as hyperconvex and hyperregular pospaces in which every monotone net has a supremum to which it converges. Moreover, we find extensions to the non-sober case by replacing suprema with cuts, and we provide topological generalizations of known facts for quasicontinuous posets. Similar results are obtained for hypercompactly based spaces and quasialgebraic posets. Furthermore, locally hypercompact spaces are described by certain relations between finite sets and points, providing a quasiuniform approach to such spaces. Our results lead to diverse old and new equivalences and dualities for categories of locally hypercompact spaces or quasicontinuous posets.

Characterizations of Various Continuities of Posets Via Approximated Elements

In this paper, various continuities of posets which may not be dcpos are considered. The concepts of approximated elements and hyper-approximated elements on posets are introduced. New characterizations of continuous posets and hypercontinuous posets are given. Meanwhile, as a generalization of approximated elements, the concept of quasi-approximated elements on dcpos is introduced and some characterizations of quasicontinuous domains are also obtained. It is proved that under some reasonable conditions, the set B(L) (resp., QB(L)) of approximated elements (resp., quasi-approximated elements) in the induced order of a dcpo L is a continuous domain (resp., a quasicontinuous domain).

Consistent Smyth powerdomains of topological spaces and quasicontinuous domains

The consistent Smyth powerdomain RQC(X) of a topological space X means the family of all nonempty relatively compact-connected saturated subsets of X, ordered by the reverse inclusion and endowed with the upper Vietoris topology. In this paper, we study properties of consistent Smyth powerdomains of certain topological spaces (especially, quasicontinuous domains equipped with the Scott topology) from topological, order theoretical and categorical aspects. Main results are: (i) a topological space X is sober iff RQC(X) is sober; (ii) if X is locally compact-connected, well-filtered and coherent, then RQC(X) is coherent; (iii) every dcpo equipped with the Scott topology is locally connected, and if a dcpo L is finitely up-generated, locally compact, well-filtered and coherent, then RQC(L) is a Lawson compact L-domain; (iv) if L is a quasicontinuous domain (resp., a Lawson compact quasicontinuous domain, a quasialgebraic domain), then RQC(L) is a continuous dcpo-∧↑-semilattice (resp., a Lawson compact L-domain, an algebraic dcpo-∧↑-semilattice); (v) it is proved that RQC(L) is a free continuous dcpo-∧↑-semilattice over a quasicontinuous domain L.

The σ1-topology and λ1-topology on s1-quasicontinuous posets

In this paper, we consider a common generalization of both s1-continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s1-quasicontinuous posets. The main results are: (1) A poset is an s1-quasicontinuous poset iff the σ1-topology is a hypercontinuous lattice iff the S⁎-convergence is topological with respect to the σ1-topology; (2) A poset is s1-continuous iff it is meet s1-continuous and s1-quasicontinuous; (3) The λ1-topology on an s1-quasicontinuous poset is Tychonoff; (4) A poset P is s1-quasicontinuous and the σ1-topology is sober iff P is a quasicontinuous domain and the σ1-topology coincides with the Scott topology.

All cartesian closed categories of quasicontinuous domains consist of domains

Quasicontinuity is a generalisation of Scott's notion of continuous domain, introduced in the early 80s by Gierz, Lawson and Stralka. In this paper we ask which cartesian closed full subcategories exist in qCONT, the category of all quasicontinuous domains and Scott-continuous functions. The surprising, and perhaps disappointing, answer turns out to be that all such subcategories consist entirely of continuous domains. In other words, there are no new cartesian closed full subcategories in qCONT beyond those already known to exist in CONT.To prove this, we reduce the notion of meet-continuity for dcpos to one which only involves well-ordered chains. This allows us to characterise meet-continuity by “forbidden substructures”. We then show that each forbidden substructure has a non-quasicontinuous function space.

Formula omitted]-Quasicontinuous posets

In this paper, we consider a common generalization of both s2-continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s2-quasicontinuous posets. The main results are: (1) The way below relation on an s2-quasicontinuous poset has the interpolation property; (2) The λ2-topology on an s2-quasicontinuous poset is completely regular; (3) A poset is s2-continuous iff it is meet s2-continuous and s2-quasicontinuous.

QFS-domains and quasicontinuous domains

In this paper, we show that (1) for each QFS-domain L, L is an ωQFS-domain iff L has a countable base for the Scott topology; (2) the Scott-continuous retracts of QFS-domains are QFS-domains; (3) for a quasicontinuous domain L, L is Lawson compact iff L is a finitely generated upper set and for any x1, x2 ∈ L and finite G1, G2 ⊆ L with G1 ≪ x1, G2 ≪ x2, there is a finite subset F ⊆ L such that ↑ x1∩ ↑ x2 ⊆↑ F ⊆↑ G1∩ ↑ G2; (4) L is a QFS-domain iff L is a quasicontinuous domain and given any finitely many pairs {(Fi, xi): Fi is finite, xi ∈ L with Fi ≪ xi, 1 ≤ i ≤ n}, there is a quasi-finitely separating function δ on L such that Fi ≪ δ(xi) ≪ xi.

The Equivalence of QRB, QFS, and Compactness for Quasicontinuous Domains

In this article we show the equivalence of QRB, QFS, and compact quasicontinuous domains. QRB and QFS domains are generalizations of RB and FS domains to the setting of quasicontinuous domains and compactness means compactness in the Lawson topology. This equivalence extends in the algebraic setting to a quasicontinuous version of bifinite domains. The Smyth powerdomain is a basic tool in the proofs, and it is shown that quasicontinuous properties at the dcpo level typically have the corresponding continuous domain properties at the Smyth powerdomain level.

Evolution of microwave ferromagnetic resonance with magnetic domain structure in FeCoBSi antidot arrays

Magnetic domain structure of FeCoBSi antidot array thin films of varying thickness were characterized using surface magneto-optic Kerr effect. Vibrating sample magnetometry and microstrip transmission line measurements helped to associate the microwave magnetic analysis of the antidot arrays with hysteresis studies. The domain structure evolution from quasi-continuous domains to strip domains induced by the competing exchange and dipolar interaction resulted in the change of ferromagnetic resonance from multi-band to single-band. Hence, the mechanisms of multi-resonance are proposed to be related to domain wall motion, natural resonance and spin wave modes. This phenomenon can be used to control the magnetization dynamics in spin wave devices.

A Note on Quasicontinuous Domains

In this note, the quasicontinuity of a dcpo and its principal ideals/filters are considered. It is shown that every principal ideal, as a subposet, in a quasicontinuous domain is also a quasicontinuous domain. A dcpo is quasicontinuous if every principal ideal is quasicontinuous and it has finitely many maximal elements. For a bc-domain being quasicontinuous, the requirement of directedness of the approximation family of each element can be omitted.

Quasicontinuous Domains and the Smyth Powerdomain

In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. This is in contrast to the previous approaches where the properties of a quasicontinuous domain were compared primarily with the properties of the lattice of Scott-open subsets. We present a characterization of those domains that occur as domains of nonempty compact saturated subsets of a quasicontinuous domain.A set theoretical lemma due to M. E. Rudin has played a crucial role in the development of quasicontinuous domains. We present a topological variant of Rudinʼs Lemma where irreducible sets replace directed sets. The notion of irreducibility here is that of a nonempty set that cannot be covered by two closed sets except if already one of the sets is covering it. Since directed sets are the irreducible sets for the Alexandroff topology on a partially ordered set, this is a natural generalization. It allows a remarkable characterization of sober spaces.For this we denote by QX the space of nonempty compact saturated subsets (with the upper Vietoris topology) of a topological space X. The following properties are equivalent: (1) X is sober, (2) QX is sober, (3) X is strongly well-filtered in the following sense: Whenever A is an irreducible subset of QX and U an open subset of X such that ⋂A⊆U, then K⊆U for some K∈A. This result fills a gap in the existing literature.

QFS-Domains and their Lawson Compactness

In this paper, concepts of quasi-finitely separating maps and quasi-approximate identities are introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are: (1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains, are all QFS-domains; (2) QFS-domains are compact in the Lawson topology; (3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is compact in the Lawson topology; (4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains.

Quasicontinuity of Posets via Scott Topology and Sobrification

In this paper, posets which may not be dcpos are considered. In terms of the Scott topology on posets, the new concept of quasicontinuous posets is introduced. Some properties and characterizations of quasicontinuous posets are examined. The main results are: (1) a poset is quasicontinuous iff the lattice of all Scott open sets is a hypercontinuous lattice; (2) the directed completions of quasicontinuous posets are quasicontinuous domains; (3) A poset is continuous iff it is quasicontinuous and meet continuous, generalizing the relevant result for dcpos.