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A proximity approach to some region-based theories of space. (English) Zbl 1185.68682

Summary: This paper is a continuation of [D. Vakarelov, I. Düntsch and B. Bennett, “A note on proximity spaces and connection based mereology”, in: C. Welty and B. Smith (eds.), Proceedings of the 2nd international conference on formal ontology in information systems (FOIS’01), 139–150 (2001)]. 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 [P. Roeper, J. Philos. Log. 26, No. 3, 251–309 (1997; Zbl 0873.54001)]. The similarity between the local proximity spaces of S. Leader [Math. Ann. 169, 275–281 (1967; Zbl 0144.21702)] and local connection algebras is emphasized. Machinery, analogous to that introduced by V. A. Efremovich [Dokl. Akad. Nauk SSSR, n. Ser. 76, 341–343 (1951; Zbl 0042.16703)], Yu. M. Smirnov [Dokl. Akad. Nauk SSSR, n. Ser. 84, 895–898 (1952; Zbl 0046.16304); Mat. Sb., N. Ser. 31 (73), 543–574 (1952; Zbl 0047.41903)] and Leader [loc. cit.] 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 [loc. cit.]. Finally, the notion of MVD-algebra is introduced. It is similar to T. Mormann’s notion of enriched Boolean algebra [Log. Log. Philos. 6, 35–54 (1998; Zbl 0974.03011)], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.

MSC:

68T27 Logic in artificial intelligence
03B30 Foundations of classical theories (including reverse mathematics)
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
54E05 Proximity structures and generalizations
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References:

[1] BENNETT B., Proceedings of ECAI-2000 pp 204–
[2] BENNETT B., Fundamenta Informaticæ 46 (1) pp 145– (2001)
[3] DOI: 10.1305/ndjfl/1093635748 · Zbl 0749.06004 · doi:10.1305/ndjfl/1093635748
[4] DOI: 10.1305/ndjfl/1039886519 · Zbl 0877.51006 · doi:10.1305/ndjfl/1039886519
[5] Č ECH E., Topological Spaces (1966) · Zbl 0141.39401
[6] CHOQUET G., Comptes-Rendus de l’Académie des Sciences 224 pp 171– (1947)
[7] DOI: 10.1305/ndjfl/1093883455 · Zbl 0438.03032 · doi:10.1305/ndjfl/1093883455
[8] DOI: 10.1305/ndjfl/1093870761 · Zbl 0597.03005 · doi:10.1305/ndjfl/1093870761
[9] COHN A., Fundamenta Informaticæ 46 pp 1– (2001)
[10] DENEVA A., Fundamenta Informaticæ 31 pp 295– (1997)
[11] DÜNTSCH I., Fundamenta Mathematicæ 39 pp 229– (1999)
[12] DÜNTSCH I., Fundamenta Informaticæ 46 pp 71– (2001)
[13] DOI: 10.1023/A:1013892110192 · Zbl 0993.03086 · doi:10.1023/A:1013892110192
[14] DÜNTSCH I., Theoretical Computer Science 255 (63) (2001)
[15] EFREMOVIČ V., DAN SSSR 76 pp 341– (1951)
[16] EFREMOVIČ V., Mat Sbornik (New Series) 31 pp 189– (1952)
[17] ENGELKING R., General Topology (1977)
[18] DOI: 10.1016/B978-044488355-1/50020-7 · doi:10.1016/B978-044488355-1/50020-7
[19] DOI: 10.1007/BF00485101 · Zbl 0201.32104 · doi:10.1007/BF00485101
[20] HAYES P. J., Formal Theories of the Commonsense World (1984)
[21] HU S.-T., J. Math. Pures Appl. 28 pp 287– (1949)
[22] DOI: 10.1007/BF01428196 · Zbl 0258.06010 · doi:10.1007/BF01428196
[23] KOPPELBERG S., Handbook on Boolean Algebras 1 (1989) · Zbl 0676.06019
[24] DOI: 10.2307/2939504 · doi:10.2307/2939504
[25] DOI: 10.1007/BF01362349 · Zbl 0144.21702 · doi:10.1007/BF01362349
[26] DOI: 10.2307/2266169 · Zbl 0023.28903 · doi:10.2307/2266169
[27] LE ŚNIEWSKI S., Prace Polskiego Kola Naukowe w Moskwie, Sekcya matematycznoprzyrodnicza 2 (1916)
[28] DOI: 10.12775/LLP.1998.002 · Zbl 0974.03011 · doi:10.12775/LLP.1998.002
[29] NAIMPALLY S. A., Proximity Spaces (1970) · Zbl 0206.24601
[30] DOI: 10.1007/3-540-55602-8_225 · doi:10.1007/3-540-55602-8_225
[31] DOI: 10.1023/A:1017904631349 · Zbl 0873.54001 · doi:10.1023/A:1017904631349
[32] SIKORSKI R., Boolean Algebras (1964)
[33] SMIRNOV J. M., Mat. Sb. 31 pp 543– (1952)
[34] STELL J., Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT 97) pp 163–
[35] DOI: 10.1016/S0004-3702(00)00045-X · Zbl 0948.68142 · doi:10.1016/S0004-3702(00)00045-X
[36] ŠWARTZ A. S., Ucen. Zap. Ivanovsk Gos. Ped. Inst. 10 pp 55– (1956)
[37] TARSKI A., First Polish Mathematical Congress, Lwø’w (1927)
[38] DOI: 10.1007/BF01431527 · Zbl 0247.54027 · doi:10.1007/BF01431527
[39] VAKARELOV D., Incomplete Information – Rough Set Analysis pp 492– (1997)
[40] VAKARELOV D., Proceedings of the 11th Amsterdam Colloquium pp 301– (1997)
[41] VAKARELOV D., Proceedings of the 2nd International Conference on Formal Ontology in Information Systems (FOIS’01) pp 139–
[42] DOI: 10.1016/S0169-023X(96)00017-1 · Zbl 0875.68388 · doi:10.1016/S0169-023X(96)00017-1
[43] WHITEHEAD A. N., Process and reality (1929) · JFM 55.0035.03
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