Continuous Semilattices Research Articles

A note on information systems for continuous semi-lattices

Wang et al. recently provided a representation for continuous semi-lattices by SL-information systems and raised a question in the conclusion section of [15]. That is, whether a modification of the notion of an approximable mapping can be made so that an equivalence to the category of continuous semi-lattices is obtained. This category has Scott-continuous functions that respect non-empty finite meets as morphisms. In this note, we solve this problem by introducing the concept of SL-approximable mappings between SL-information systems. As an application, we propose the BC-approximable mappings between BC-information systems and establish a categorical equivalence between BC-information systems with BC-approximable mappings and bounded-complete domains with Scott-continuous functions that preserve non-empty infima.

Information systems for continuous semi-lattices

We consider two problems on representation theory for domains by information systems. The first problem is how to represent continuous semi-lattices as information systems, and the second one is which subclass of the continuous information systems designed by Spreen et al. for continuous domains can be used to represent bounded complete domains. We solve the first problem by defining a notion of SL-information systems. This is a special case of the continuous information systems. We also discuss the morphism part of SL-information systems, which corresponds to Scott-continuous functions between continuous semi-lattices. Noting that each bounded complete domain is a special continuous semi-lattices, we provide a correct solution to the second problem by properly strengthening the conditions of SL-information systems.

Continuous Domains in Formal Concept Analysis*

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

Compatibilities between continuous semilattices

We define compatibilities between continuous semilattices as Scott continuous functions from their pairwise cartesian products to $\{0,1\}$ that are zero preserving in each variable. It is shown that many specific kinds of mathematical objects can be regarded as compatibilities, among them monotonic predicates, Galois connections, completely distributive lattices, isotone mappings with images being chains, semilattice morphisms etc. Compatibility between compatibilities is also introduced, it is shown that conjugation of non-additive real-valued or lattice valued measures is its particular case.

Representations of stably continuous semi-lattices by information systems and abstract bases

In this paper, we introduce two new notions of S-continuous information systems and S-abstract bases which provide concrete representations of stably continuous semi-lattices. We shed light on how the states of an S-continuous information system and the round ideal completion of an S-abstract basis form a stably continuous semi-lattice, respectively. Then we obtain two representation theorems for stably continuous semi-lattices. What is more, we show the equivalence of the categories corresponding to these three structures.

Rough games modeled via L-fuzzy ambiguous representations of semilattices

Monotonic predicates and L-fuzzy ambiguous representations of continuous semilattices with bottom elements are applied to describe rough normal-form games and rough extensive-form two-player zero-sum games with imperfect information. Payoff predicates are introduced to find best guaranteed gains of the players. These predicates are calculated recursively via backward payoff predicate transformers, which are also studied. It is shown how payoff predicates determine almost optimal strategies. An approximate minimax theorem is also stated.

Ambiguous Representations of Semilattices, Imperfect Information, and Predicate Transformers

Crisp and lattice-valued ambiguous representations of one continuous semilattice in another one are introduced and operation of taking pseudo-inverse of the above relations is defined. It is shown that continuous semilattices and their ambiguous representations, for which taking pseudo-inverse is involutive, form categories. Self-dualities and contravariant equivalences for these categories are obtained. Possible interpretations and applications to processing of imperfect information are discussed.

Universal Approach to Z–frame Envelopes of Semilattices

In this paper we deal with systems of sets on the category of all semilattices. Based on the result of Z–frame freely generated by Z–sites studied in [D. Zhao, “Generalization of Frames and Continuous Lattices,” Ph.D. thesis, Cambridge University, 1993], we prove that every semilattice admits a Z–frame envelope. It is a universal approach for any system Z of sets. We define and study a new continuity named Za–continuty on semilattice and prove that in some specific systems of sets (Low, Fin and Idl), every Za–continuous semilattice admits a Z–continuous envelope.

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F -distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.

Compatible algebraic structures on scattered compacta

It is proved that each hereditarily collectionwise Hausdorff compact scattered space with finite scattered height admits a continuous semilattice operation turning it into a topological semilattice with open principal filters. On the other hand a compactification γ N of a countable discrete space N whose remainder is homeomorphic to [ 0 , ω 1 ] admits no (separately) continuous binary operation turning γ N into an inverse semigroup (semilattice). Also we construct a compactification ψ N of N admitting no separately continuous semilattice operation and such that the remainder ψ N ∖ N is homeomorphic to the one-point compactification of an uncountable discrete space. To show that ψ N admits no continuous semilattice operation we prove that the set of isolated points of a compact scattered topological semilattice X of scattered height 2 is sequentially dense in X. Also we prove that each separable scattered compactum with scattered height 2 is a subspace of a separable compact scattered topological semilattice with open principal filters and scattered height 2. This allows us to construct an example of a separable compact scattered topological semilattice with open principal filters and scattered height 2, which fails to be Fréchet–Urysohn. Also we construct an example of a Fréchet–Urysohn separable non-metrizable compact scattered topological semilattice with open principal filters and scattered height 2.

The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification

Let L be a continuous semilattice. We use USC(X, L) to denote the family of all lower closed sets including X x 0 in the product space X x ΛL and ↓ C(X, L) the one of the regions below of all continuous maps from X to ΛL. USC(X, L) with the Vietoris topology is a topological space and ↓ C(X, L) is its subspace. It will be proved that, if X is an infinite locally connected compactum and ΛL is an AR, then USC(X, L) is homeomorphic to [-1,1]ω. Furthermore, if L is the product of countably many intervals, then ↓ C(X, L) is homotopy dense in USC(X, L), that is, there exists a homotopy h : USC(X, L) × [0,1] → USC(X, L) such that h0 = idUSC(X, L) and ht(USC(X, L)) ⊂↓C(X, L) for any t > 0. But ↓C(X, L) is not completely metrizable.

Idempotent analysis and continuous semilattices

In this survey article we give a brief overview of various aspects of the recently emerging field of idempotent analysis and suggest potential connections with domain theory.

Existence theorems for measures on continous posets, with applications to random set theory.

We state conditions on a partially ordered set L and a mapping λ, defined on a class of filters on L, under which λ extends to a measure on the minimal σ-field over this class. By applying this result to the case when L is a continuous lattice, all locally finite measures on L are identified as well as all Levy-Khinchin measures. We then characterize these kinds of measures on continuous semilattices and continuous posets. The correspondence between probability measures on the line and distribution function is a particular case of this result. We give a simple proof of the Daniell-Kolmogorov existence theorem for probability measures on products of continuous lattices

Continuous semilattices

Continuous semilattices are semilattices endowed with an order for which the semilattice operation is continuous, i.e., preserves specified joins. For various types of specification of join-continuity, we describe the free continuous semilattices.

Infinite antichains in semilattices

In this paper we consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semilattices. Conditions are first considered that lead to the antichain generating a copy of a countable product of the two-element semilattice. Then a special semilattice, called Δ, is defined, its basic properties developed, and it is shown in our main result that a continuous semilattice with an infinite antichain converging to a larger element contains a semilattice copy of Δ. The paper closes with a consideration of countable antichains that converge to a lower element or a parallel element and the kinds of semilattices generated in this context.

Algebraic theories for continuous semilattices

Algebraic theories for continuous semilattices

Continuous lattices and compact Lawson semilattices

Continuous lattices and compact Lawson semilattices