Proximity-type Relations on Boolean Algebras. and their Connections with Topological Spaces by Georgi Dobromirov

Transcription

1 REPORT ON THE DOCTORAL DISSERTATION Proximity-type Relations on Boolean Algebras and their Connections with Topological Spaces by Georgi Dobromirov Dimov 1. Scientific contributions in the dissertation The doctoral dissertation Proximity-type Relations on Boolean Algebras and their Connections with Topological Spaces by Georgi Dobromirov Dimov is written on iv pages: Preface 1 12, 6 chapters (0. Foreword (13 40), 1. MVD-algebras and a new proof of Roepers Representation Theorem 41 72, 2. Some generalizations of de Vries and Fedorchuks Duality Theorems , 3. Some generalizations of the Stone Duality Theorem , 4. Some applications in General Topology , 5. Some Isomorphism Theorems for Scott and Tarski consequence systems ), Bibliography , 126 references. The dissertation mainly concerns generalizations and extensions of the following famous duality results: (1) [Stone duality theorem, 1937]: The category of zero-dimensional Hausdorff compact spaces and continuous mappings is dual to the category of Boolean algebras and Boolean homomorphisms. (2) [de Vries duality theorem, 1962] The category of Hausdorff compact spaces and continuous mappings is dual to the category DHC (defined by de Vries) of all complete normal contact algebras and the appropriate morphisms. (3) [Fedorchuk duality theorem, 1973] The category QHC of Hausdorff compact spaces and quasi-open mappings is dual to the subcategory DQHC of DHC of all complete compingent Boolean algebras and complete DHCmorphisms. and the equivalence theorem (4) [Fedorchuk equivalence theorem, 1973] The category QHC is equivalent to the category EQHC of complete normal contact Boolean algebras and appropriate morphisms (quite different from the DQHC-morphisms). Generalizations of these results are usually done for locally compact Hausdorff spaces and (different sorts of) continuous mappings between them. Then, these results are applied to a few fields of mathematics. 1

2 This dissertation contains many nice, significant results and their applications, and I am going to demonstrate the importance of these results. Chapter 0 contains, as usual, terminology and notation, basic notions and results which be used in the dissertation. The central result of Chapter 1, which is a basic result for author s study in this dissertation, is a new proof of the Roeper representation theorem (and its special case de Vries representation theorem from 1962) for normal contact (Boolean) algebras. In 1997, P. Roeper introduced the notion of region-based topology, a Boolean algebra with a contact relation satisfying certain axioms and a predicate of limitedness, and proved that there is a bijective correspondence between the class of all (up to isomorphism) region-based topologies and the class of all (up to homeomorphism) locally compact Hausdorff spaces. The author of this dissertation realized that there is a big similarity between axioms of region-based topology and Leader s axioms of local proximity spaces introduced by Leader in This is why the author uses a new name local contact algebras (this notion is equivalent to the notion of MDV-algebra introduced by the author and his coauthors) for region-based topologies. Applying methods of local proximity spaces which are essentially lattice-theoretic versions of the proximity methods and employing some new ideas the author gives a shorter proof of Roeper s theorem. The methods and constructions in this new exposition of Roepers Representation Theorem allow to the author to apply them several times for proofs of other duality results. Using the Stone Duality Theorem for Boolean algebras and some ideas from Leaders proof of the Smirnov Compactification Theorem, the author provides a new proof of de Vries representation theorem for normal contact algebras. Chapter 2 contains generalizations of de Vries duality theorem, Fedorchuk s duality theorem, and Fedorchuk s equivalence theorem, as well as several other new theorems on duality and equivalence. The central result in this chapter is Theorem and its nice proof. The author found the category DHLC which is dual to the category HLC of Hausdorff locally compact spaces and continuous mappings: the dual object to a locally compact Hausdorff space X is the local contact (Boolean) algebra (RC(X), ρ X, CR(X)), where RC(X) is the Boolean algebra of regular closed subsets of X, ρ X is the relation on RC(X) defined by Aρ X B A B, and CR(X) is the family/ideal of compact regular closed subsets of X. It is also proved a similar theorem for the category of connected locally compact 2

3 Hausdorff spaces. Good applications of the technique in the proof of Theorem are direct descriptions of DHLC-products of arbitrary families of complete local contact algebras (Subsection 2.3.2), DHLC-sums of finite families of complete local contact algebras and DHC-sums of arbitrary families of complete normal contact algebras (Subsection 2.3.3), and the existence and uniqueness of completion of local contact algebras (Subsection 2.3.4). In Subsection 2.4, using Theorem , the author proves a generalization of de Vries duality theorem to the category of locally compact Hausdorff spaces and perfect mappings (Theorem ; the author gives two nice proofs of this theorem, and derives some its consequences). Fedorchuk s duality theorem for the category QHC is generalized to the category SHLC of locally compact Hausdorff spaces and skeletal mappings (for every open V Y, int(f Cl(V )) Cl(f (V ))) between them (Theorem ), and also to the subclass PSHLC of SHLC of all locally compact Hausdorff spaces and skeletal perfect mappings (Theorem ). Let me mention that Theorems , and present new (even in the class of compact Hausdorff spaces) duality results related to compact Hausdorff spaces and open mappings between them. Fedorchuk s equivalence theorem concerns the category QHC of compact Hausdorff spaces and quasi-open mappings between them (f : X Y is quasi-open if for each non-empty open set U X the set intf(u) is non-empty). Theorems and give two generalizations of Fedorchuk s result to the categories SHLC and PSHLC, respectively. Some new interesting equivalence theorems are also presented (Theorems , and ). In Chapter 3, the celebrated Stone duality theorem is generalized to the category BoolSp of zero-dimensional locally compact Hausdorff spaces and continuous mappings. Some duality results for certain subcategories of the category BoolSp are also shown. Theorems , , , , , are new even in the case of compact Hausdorff spaces. The main notions used in these proofs are the notions of local Boolean algebra and prime local Boolean algebra introduced by the author of this dissertation. It is proved that BoolSp and the class of local Boolean algebras are contravariant structures (Theorem ). Theorem describes the category ZLBA of special local Boolean algebras which is dually equivalent to the category BoolSp. The category of compact zero-dimensional Hausdorff spaces and quasi-open maps is dually equivalent to the category of Boolean algebras and complete Boolean homomorphisms (Corollary ); 3

4 the category of compact zero-dimensional Hausdorff spaces and open maps is dually equivalent to the category of Boolean algebras and Boolean homomorphisms of a special kind (Corollary ). Chapter 4 contains results concerning applications of the results obtained in the previous chapters to topological spaces. We describe the main results of this chapter. In order to describe the ordered set (LC, ) of all Hausdorff local compactifications of a Tychonoff space X only by proximities, the author defines the category LCP of lc-proximity spaces and lc-mappings and proves that this category is equivalent to the category LP, introduced by Leader in 1967, of separate local proximity spaces and equivariant mappings between them. Considering the important problem of extensions of continuous mappings between Tychonoff spaces and using results from Chapter 2, in particular Theorem , the author proves a series of theorems related to this problem. Theorems and are generalizations of the results by Polyakov and Leader and characterize mappings between Tychonoff spaces which have a continuous extension to an arbitrary but fixed Hausdorff locally compact extensions of these spaces. Other results are related to different sorts of continuous mappings: open, perfect, quasi-open, skeletal. Results from Chapter 3, especially Theorem , are applied to generalize some results of B. Banaschewski, G. Bezhanishvili, and Ph. Dwinger in connection with zero-dimensional locally compact extensions of spaces. Subsection 4.4 contains nice results about objects dual to the real line, the closed unit interval, Euklidean spaces R n, spheres S n, and the Tychonoff cube I κ and the tori T κ, κ a cardinal number. These results are obtained by applying results from Chapter 2. Another applications of results in Chapters 2 and 3 are described in Chapter 5 and concern the notions of Scott consequence systems (Ssystems) introduced by D. Vakarelov in 1992 (and in a similar form by D. Scott in 1982), and Tarski consequence systems, briefly T -systems (Tarski, 1955; Vakarelov, 1992). The category SSyst of S-systems and S-morphisms is defined, and also its (full) subcategory TSyst of all Tarski consequence systems. A result by Vakarelov is extended, a result by Prodanov is generalized and a series of isomorphism theorems are proved. For example, it is shown that the categories BoolAlg of all Boolean algebras and all Boolean homomorphisms and DLat of all distributive lattices and lattice homomorphisms are isomorphic and are both isomorphic to certain (reflective full) subcategories of the category SSyst (Proposition , Theorem , Corollary ). The category TSyst is isomorphic to a subcategory of 4

5 the category Top of topological spaces and continuous mappings, where objects of the subcategory are certain hyperspaces (Theorem ). It is also shown that the category SSyst is isomorphic to a full subcategory of the category TPS of so-called topological property systems, introduced in this dissertation (Theorem ). Author s papers cited in the dissertation The list of (126) references contains relevant literature related to the theme considered in this dissertation. We like to discuss citations of papers by the author of the dissertation included in it. We use the enumeration from the dissertation. Author s papers included in the list of references are the papers [25 45, 118], but the dissertation is written mainly on the basis of the papers [27 34, 39, 42, 118]; some of these papers have been written jointly with coauthors (the papers [35] [45] and [118]). We list here the papers published in journals with impact factor (IF): [27] Applied Categorical Structures (2009), IF 0,548 [28] Topology and its Applications (2009), IF [29] Acta Mathematica Hungarica (2010), IF [30] Topology and its Applications (2010), IF [31] Acta Mathematica Hungarica (2011), IF [33] Publicationes Mathematicae Debrecen (2012), IF [43] Fundamenta Informaticae (2006), IF [44] Fundamenta Informaticae (2006), IF Papers [32] (Questions and Answers in General Topology, 2011), [39] (Fundamenta Informaticae, 1998) and [118] (Journal of Applied Non-Classical Logics, 2002) are published in high quality journals, unfortunately, without IF in years of their publications. Contributions of Georgi Dimov in joint papers are equal to contributions of his coauthors. Citations of author s papers The importance of the results in the dissertation can be seen from the list of citations of the papers on the basis of which this dissertation is written. This list contains 78 citations, at least 13 in journals with IF, without selfcitations and citations of joint papers by Dimov s coauthors. 5

6 Here is the list of citations (by the end of May, 2012). The enumeration is again taken from Bibliography in the dissertation. The paper [27] is quoted once [28] is quoted once [39] is quoted 3 times [43] is quoted 33 times [44] is quoted 18 times [45] is quoted 4 times [118] is quoted 18 times The total number of citations of Dimov s papers is over 130. About author s report on the dissertation Author s review on the dissertation very well and precisely reflects the results presented in this work. Critical observations by the referee I do not have such remarks and comments. Conclusion According to the above it is clear that Georgi Dobromirov Dimov fulfills all the conditions for the degree of doctor of science in the field of Mathematics. It is also my personal opinion based on my knowledge of Dimov s scientific and professional work. He solved some important mathematical problems, generalized and extended fundamental theorems on duality and equivalence, obtained series of new results of this sort, developed original new techniques applicable in other branches of mathematics. His papers are published in prestigious mathematical journals and presented on international mathematical conferences, and are well accepted among mathematicians as the number of citations of his papers show. Georgi Dimov has my strongest recommendation for the oral defense of his doctoral dissertation for the degree of doctor of science. Prof. Ljubiša D.R. Kočinac University of Niš Serbia lkocinac@gmail.com 6