Boolean ?-algebras and quasiopen mappings

Fedorchuk, V. V.

doi:10.1007/bf00969913

Cited by 22 publications

(58 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is an LCA-isomorphism (see (11)). We will show that λ g : (15) and (16), it is enough to show that λ g…”

Section: The Definition Of ξ Tmentioning

confidence: 99%

“…Let us mention that, using de Vries Duality Theorem, V. V. Fedorchuk [11] showed that the category of all compact Hausdorff spaces and all quasiopen maps between them is dually equivalent to the category of all complete compingent Boolean algebras and all complete Boolean homomorphisms between them satisfying one simple condition, and that in [7,8] some extensions of the Fedorchuk Duality Theorem ( [11]) to some categories whose objects are all locally compact Hausdorff spaces are obtained.…”

Section: Introductionmentioning

confidence: 99%

“…A contact algebra (B, C) is called a normal contact algebra (abbreviated as NCA) ( [6,11]) if it satisfies the following axioms (we will write " − C" for "not C"): (C5) If a(−C)b then a(−C)c and b(−C)c * for some c ∈ B; (C6) If a = 1 then there exists b = 0 such that b(−C)a. A normal CA is called a complete normal contact algebra (abbreviated as CNCA) if it is a CCA.…”

Section: Introductionmentioning

confidence: 99%

“…A normal CA is called a complete normal contact algebra (abbreviated as CNCA) if it is a CCA. The notion of normal contact algebra was introduced by Fedorchuk [11] under the name Boolean δ-algebra as an equivalent expression of the notion of compingent Boolean algebra of de Vries. We call such algebras "normal contact algebras" because they form a subclass of the class of contact algebras.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Generalization of De Vries Duality Theorem

Dimov

2008

Appl Categor Struct

View full text Add to dashboard Cite

show abstract

“…is an LCA-isomorphism (see (11)). We will show that λ g : (15) and (16), it is enough to show that λ g…”

Section: The Definition Of ξ Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Generalization of De Vries Duality Theorem

Dimov

2008

Appl Categor Struct

View full text Add to dashboard Cite

show abstract

“…In 1973, Fedorchuk [12] notes that the complete DVAL-morphisms (= DVAL-morphisms which are complete Boolean homomorphisms) have a very simple description and, moreover, the DVAL-composition of two such morphisms coincides with their settheoretic composition. He considers the category DSkeC of complete compingent Boolean algebras and complete DVALmorphisms.…”

Section: Introductionmentioning

confidence: 98%

Some generalizations of Fedorchuk duality theorem—I

Dimov

2009

Topology and its Applications

View full text Add to dashboard Cite

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. IntroductionIn the first quarter of 20th century, de Laguna [4] and Whitehead [30] initiate a new theory of space which in nowadays is well developed and is known as region-based theory of space. It is an alternative to the classical point-based theory of space and concerns not only R 3 , as it was in the very beginning, but also some large classes of topological spaces. The main ideas of the region-based theory of space can be formulated as follows:• the notion of point is too abstract to be taken as a primitive notion of the theory of space; instead of points, some more realistic spatial entities have to be put as primitives on the basis of the theory of space-in de Laguna [4] they are called solids and Whitehead [30] calls them regions;• some basic relations (like part-of, overlap, contact (or connection, in Whitehead's terminology)) between the regions has to be considered;• the points must not be disregarded; they have to be defined by means of the regions and some of the basic relations between them;• an equivalence (in some sense) between the region-based approach and the point-based approach has to be obtained. Topology and its Applications 156 (2009) 728-746 729 The notion of regular closed (or regular open) subset of a topological space is considered as a standard point-based model of the notion of region; in this model, two regular closed sets (i.e., regions) are in contact iff they have a non-empty intersection.De Laguna and Whitehead do not present their ideas in the form of a rigorous mathematical theory. This is done later by some other authors. A good survey of the region-based approach to the theory of space is given by Gerla in [17]. Recent surveys in this field, pointing out its relations with Theoretical Computer Science and AI, can be found in [2].The ideas of de Laguna and Whitehead lead naturally to the following general programme:• define in topological terms those subsets of a topological space that correspond most closely to the idea of regions;• choose some (algebraic) structure which is inherent to the family of all regions of a topological space, fix some kind of morphisms between the obtained (algebraic) objects and build in this way a category A;• find a subcategory T of the catego...

show abstract

Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics

Vakarelov

Mathematical Problems From Applied Logic II

View full text Add to dashboard Cite

Boolean ?-algebras and quasiopen mappings

Cited by 22 publications

References 3 publications

A Generalization of De Vries Duality Theorem

A Generalization of De Vries Duality Theorem

Some generalizations of Fedorchuk duality theorem—I

Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics

Contact Info

Product

Resources

About