Algebraic L-domains Research Articles

A set-theoretic approach to algebraic L-domains

AbstractIn this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.

A direct approach to representing algebraic domains by formal contexts

This paper is to establish closer links between domain theory and Formal Concept Analysis (FCA). We propose the notion of an optimised concept for a formal context, which has some properties similar to an intent. With the tool of optimised concepts, we show that the class of formal contexts has directly corresponded with algebraic domains. Meanwhile, two subclasses of formal contexts are identified to characterize algebraic L-domains and Scott domains. As an application, we resolve the open problem of how to reconstruct bounded complete continuous domains in the languages of attribute continuous contexts. Finally, we extend our presentation of algebraic domains to a categorical equivalence.

Topological representations of Lawson compact algebraic L-domains and Scott domains

Topological representations of Lawson compact algebraic L-domains and Scott domains

New representations of algebraic domains and algebraic L-domains via closure systems

New representations of algebraic domains and algebraic L-domains via closure systems

Generalised information systems capture L-domains

A generalisation of Scott's information systems [15] is presented that captures exactly all L-domains. The global consistency predicate in Scott's definition is relativised in such a way that there is a consistency predicate for each atomic proposition (token) saying which finite sets of such statements express information that is consistent with the given statement.It is shown that the states of such generalised information systems form an L-domain, and that each L-domain can be generated in this way, up to isomorphism. Moreover, the equivalence of the category of generalised information systems with the category of L-domains is derived. In addition, it will be seen that from every generalised information system capturing an algebraic bounded-complete domain a corresponding Scott information system can be obtained in an easy and natural way, and vice versa; similarly for Hoofman's continuous information systems [9] and the continuous bounded-complete domains captured by them; for Chen and Jung's disjunctive propositional logic [4] and algebraic L-domains (as well as for Wang and Li's [21] finitary version and Lawson-compact algebraic L-domains); and for Wang and Li's conjunctive sequent calculi [20] and proper continuous bounded-complete domains. The proofs always contain syntactic translations between the logical calculi involved.

A logic for Lawson compact algebraic L-domains

In this paper, we build a logic which is named N-sequent calculus. Based on this logic, we provide two kinds of logical representations of Lawson compact algebraic L-domains: one in terms of logical algebras and the other in terms of logical syntax.The first representation takes the corresponding logical algebras as research objects. The use of prime filters achieves the connection between our logic and Lawson compact algebraic L-domains. This approach is inspired by Abramsky's SFP domain logic and the disjunctive propositional logic on algebraic L-domains introduced by Yixiang Chen and Achim Jung. However, there are essential differences between them at the morphisms part. For the second representation, we directly adopt N-sequent calculi themselves as objects instead of the logical algebras. Then we establish the category of N-sequent calculi with consequence relations equivalent to that of Lawson compact algebraic L-domains with Scott continuous maps. This demonstrates the capability of the syntax of the logic in representing domains.

Locally complete consistent F-augmented contexts: A category-theoretic representation of algebraic L-domains

In this paper, we demonstrate the capability of formal concept analysis in representing a special partially ordered structures named algebraic L-domains. In particular, we introduce the notion of locally complete consistent F-augmented contexts and prove that its associated category with F-approximable connections as morphisms is precisely equivalent to that of algebraic L-domains with Scott continuous functions as morphisms. This result provides a concrete representation of algebraic L-domains via formal concept analysis as an efficient approach.

Cartesian Closed Subcategories of CON T ≪ ∗ $CONT_{\ll }^{{\kern 8pt}\ast }$

Let CONT ≪ be the category of continuous domains and Scott continuous mappings that preserve the way-below relation on domains. Let ω-ALG ≪ be the full subcategory of CONT ≪ consisting of all countably based algebraic domains, and F I N be the category of finite posets and monotone mappings. The main result proved in this paper is that F I N is the largest Cartesian closed full subcategory of ω-ALG ≪. On the other hand, it is shown that the algebraic L-domains form a Cartesian closed full subcategory of ALG ≪.

Representations of Algebraic Domains and Algebraic L-domains by Information Systems

Information systems play an important role in characterizing order structures. In this paper, we introduce the notions of the algebraic information system and algebraic L-information system. They are of the same logic-oriented style as the information system introduced by Scott (1982). But the axioms in this paper are briefer than reported in existing work. We also prove that the two new information systems exactly represent the algebraic domains and algebraic L-domains respectively. Based on the notion of approximable mapping between the algebraic information systems and the algebraic L-information systems, we obtain the result that the corresponding categories of algebraic information systems and algebraic L-information systems are equivalent to the category of algebraic domains and algebraic L-domains respectively.

Formal Contexts for Algebraic Domains

In this paper, we investigate the representation of algebraic domains by means of Formal Concept Analysis. For a formal context, we can define a large number of consistent sets. Associated with each consistent set, there is a set of F-approximable concepts which are selected from the well known approximable concepts. By virtue of F-approximable concepts, formal contexts and algebraic domains are able to interpret each other. Moreover, by analyzing the finitely consistent sets, the algebraic bifinite domains, algebraic L-domains are exactly located at the corresponding formal contexts, respectively.

Characterizations of algebraic L-domains and equivalence of relevant categories

This paper concerns algebraic L-domains. Main results are: (1) All the compact elements of an algebraic L-domain forms an L-cusl; (2) An algebraic L-domain equipped with the Scott topology is the sobrification of the set of all compact elements equipped with the Alexandrov topology; (3) The ideal completion of an L-cusl is an algebraic L-domain. (4) It is also proved that with proper concepts of L-cusl embeddings and projection pairs, the category of algebraic L-domains and projection pairs is equivalent to the category of L-cusls and L-cusl embeddings. Mathematics Subject Classification: 06A11; 18A40; 54H10

A logical approach to stable domains

Building on earlier work by Guo-Qiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic L-domains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic L-domains.

Representation theorems for directed completions of consistent algebraic L-domains

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory.

The dependent product construction in various categories of domains

Coquand, Gunter, and Winskel show how to use the class S of all (countably based) Scott domains as a model for the polymorphic lambda calculus. In particular, a type which contrains n free type variables is interpreted as a continuous functor F:S n →S. The category of continuous sections of the corresponding Grothendieck fibration is shown to be isomorphic to a Scott domain. We show that this construction does not work for bifinite domains or algebraic L-domains

Cartesian closed categories of algebraic CPOs

The results presented in this paper were inspired by the work of Smyth, who showed that there is a largest cartesian closed full subcategory inside the category of countably based algebraic cpos with least element, namely the category of profinite domains. Removing the countability condition, we show that there are exactly two maximal cartesian closed full subcategories inside the category of algebraic cpos with least element. One is the natural extension of the class of profinite domains to the uncountable case, the other is a new class of domains, which are characterized by the property that every principal ideal is a complete lattice. The name L- domain is introduced for cpos with this property. Passing to the general situation where no least element is required anymore, we find a pair of categories in place of each the profinite domains and the algebraic L-domains. Thus there are four maximal cartesian closed categories of algebraic cpos in this case. This complete overview over the possible classes of domains allows to prove general theorems about them. This is illustrated by the result that a cpo has an algebraic function space if and only if its space of strict continuous functions is algebraic.