Georgi Dimov

Abstract: This paper is the second part of the paper [2]. Both of themare in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name "Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T$_0$-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T$_0$-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.

Abstract: This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roeper's notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roeper's paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormann's notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that...

Abstract: The notions of 2-precontact and 2-contact spaces as well as of extensional (and other kinds) 3-precontact and 3-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras (satisfying some additional axioms) are proved. They incorporate and strengthen both the discrete and topological representation theorems from [3, 1, 2, 4, 10]. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. In particular, such a bijective correspondence for the RCC systems of [8] is obtained, strengthening in this way the previous representation theorems from [4, 1].

Abstract: Generalizing de Vries’ duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.

Abstract: This paper is a continuationof[8] andin it some applicationsofthe methodsand results of [8] and of [28, 7, 24, 9, 10, 11] are given. In particular,some generalizations of the Stone Duality Theorem [28] are obtained; acompletion theorem for local contact Boolean algebras is proved; a directproof of the Ponomarev’s solution [22] of Birkhoff’s Problem 72 [5] is found,and the spaces which are co-absolute with the (zero-dimensional) Eberleincompacts are described.2000 MSC: primary 18A40, 54D45; secondary 06E15, 54C10, 54E05, 06E10.Keywords: Local contact Boolean algebras; Local Boolean algebras; Prime LocalBoolean algebras; Locally compact spaces; Continuous maps; Perfect maps; Duality;Eberlein compacts; Co-absolute spaces. Introduction This paper is a second part of the paper [8]. In it we will use the notions, no-tations and results of [8] and we will apply the methods and results obtainedin [8] and in [28, 7, 24, 9, 10, 11].In Section 1, some generalizations of the Stone Duality Theorem [28]are obtained. Namely, five categories LBA, ZLBA, PZLBA, PLBA andGBPL are constructed. We show that there exists a contravariant ad-junction between the first of these categories and the category ZLC ofzero-dimensional locally compact Hausdorff spaces (=

Abstract: What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Abstract: In discussing the philosophical system put forward in PR, we need to point out that not only does Whitehead introduce a novel terminology, but the work itself is somewhat amorphous in character, and this despite his attempt to state a categoreal scheme — a general scheme of ideas in terms of which all our experience is to be described. There is also a considerable amount of overlap between the various parts of this book. It might have been a clearer and more effective work if Whitehead had engaged in some judicious pruning before publication.

Abstract: This paper is the second part of the paper [2]. Both of themare in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name "Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T$_0$-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T$_0$-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.