Geodesic
Dynamic mereotopology.~III. Whiteheadian type of integrated point-free theories of space and time.~III
Algebra i logika, Tome 55 (2016) no. 3, pp. 273-299.

Voir la notice de l'article provenant de la source Math-Net.Ru

This is the third in a three-part series of papers shortly denoted by Part I [Algebra i Logika, 53, No. 3 (2014), 300–322], Part II [Algebra i Logika, 55, No. 1 (2016), 14–36], and Part III. The papers mentioned are devoted to some Whiteheadean theories of space and time. Part I contains a historical introduction and some facts from static mereotopology. Part II introduces a point-based definition of a dynamic model of space and the definition of a standard dynamic contact algebra based on the so-called snapshot construction. The given model has an explicit time structure with an explicit set of time points equipped with a before-after relation and a set of regions changing in time, called dynamic regions. The dynamic model of space contains several definable spatio-temporal relations between dynamic regions: space contact, time contact, precedence, and some others. In Part II, a number of statements for these relations are proven, which in the present Part III are taken as axioms for the abstract definition of some natural classes of dynamic contact algebras, considered as an algebraic formalization of dynamic mereotopology. Part III deals with a representation theory for dynamic contact algebras, and the main theorem says that each dynamic contact algebra in some natural class is representable as a standard dynamic contact algebra in the same class.
Keywords: dynamic contact algebra, dynamic mereotopology, point-free theory of space and time, representation theorems. 
@article{AL_2016_55_3_a0,
     author = {D. Vakarelov},
     title = {Dynamic {mereotopology.~III.} {Whiteheadian} type of integrated point-free theories of space and {time.~III}},
     journal = {Algebra i logika},
     pages = {273--299},
     publisher = {mathdoc},
     volume = {55},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2016_55_3_a0/}
}
TY  - JOUR
AU  - D. Vakarelov
TI  - Dynamic mereotopology.~III. Whiteheadian type of integrated point-free theories of space and time.~III
JO  - Algebra i logika
PY  - 2016
SP  - 273
EP  - 299
VL  - 55
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2016_55_3_a0/
LA  - ru
ID  - AL_2016_55_3_a0
ER  - 
%0 Journal Article
%A D. Vakarelov
%T Dynamic mereotopology.~III. Whiteheadian type of integrated point-free theories of space and time.~III
%J Algebra i logika
%D 2016
%P 273-299
%V 55
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2016_55_3_a0/
%G ru
%F AL_2016_55_3_a0
D. Vakarelov. Dynamic mereotopology.~III. Whiteheadian type of integrated point-free theories of space and time.~III. Algebra i logika, Tome 55 (2016) no. 3, pp. 273-299. http://geodesic.mathdoc.fr/item/AL_2016_55_3_a0/

[1] D. Vakarelov, “Dinamicheskaya mereotopologiya. III. Edinye tochechno-svobodnye teorii prostranstva i vremeni tipa Uaitkheda. I”, Algebra i logika, 53:3 (2014), 300–322 | MR | Zbl

[2] D. Vakarelov, “Dinamicheskaya mereotopologiya. III. Edinye bestochechnye teorii prostranstva i vremeni tipa Uaitkheda. II”, Algebra i logika, 55:1 (2016), 14–36 | DOI | Zbl

[3] D. Vakarelov, “Dynamic mereotopology: A point-free theory of changing regions. I. Stable and unstable mereotopological relations”, Fundam. Inform., 100:1–4 (2010), 159–180 | MR | Zbl

[4] D. Vakarelov, “Dynamic mereotopology. II: Axiomatizing some Whiteheadian type space-time logics”, Proc. 9th conf. AiML 2012 (Copenhagen, Denmark, August 22–25, 2012), Advances in modal logic, 9, eds. Th. Bolander et al., College Publ., London, 2012, 538–558 | MR | Zbl

[5] D. Vakarelov, “A modal approach to dynamic ontology: modal mereotopology”, Log. Log. Philos., 17:1/2 (2008), 163–183 | MR | Zbl

[6] I. Düntsch, M. Winter, “Moving spaces”, Proc. 15th Int. Symp. on temporal representation and reasoning, TIME2008 (Montréal, Canada, June 16–18, 2008), eds. St. Demri, Ch. S. Jensen, IEEE Computer Society Press, Los Alamitos, 2008, 59–63

[7] I. Düntsch, M. Winter, “Timed contact algebras”, Proc. 16th Int. Symp. on temporal representation and reasoning, TIME 2009 (Brixen, Italy 23–25 July 2009), eds. C. Lutz, J-F. Raskin, IEEE Computer Society Press, Los Alamitos, 2009, 133–138

[8] V. Nenchev, “Logics for stable and unstable mereological relations”, Cent. Eur. J. Math., 9:6 (2011), 1354–1379 | DOI | MR | Zbl

[9] V. Nenchev, “Dynamic relational mereotopology: logics for stable and unstable relations”, Log. Log. Philos., 22:3 (2013), 295–325 | MR | Zbl

[10] R. Konchakov, A. Kurucz, F. Wolter, M. Zakharyaschev, “Spatial logic $+$ temporal logic $=$ ?”, Handbook of spatial logics, eds. M. Aiello et al., Springer-Verlag, Dordrecht, 2007, 497–564 | DOI | MR

[11] R. Goldblatt, “Diodorean modality in Minkowski spacetime”, Stud. Log., 39 (1980), 219–236 | DOI | MR | Zbl

[12] V. B. Shekhtman, “Modal logics of domains on the real plane”, Stud. Log., 42 (1983), 63–80 | DOI | MR | Zbl

[13] I. Shapirovsky, V. Shehtman, “Chronological modality in Minkowski spacetime”, Sel. papers the 4th conf. AiML 2002 (Toulouse, France, October 2002), Advances in modal logic, 4, eds. P. Balbiani et al., King's College Publ., London, 2003, 437–459 | MR | Zbl

[14] I. Shapirovsky, V. Shehtman, “Modal logics of regions and Minkowski spacetime”, J. Log. Comput., 15:4 (2005), 559–574 | DOI | MR | Zbl

[15] A. N. Whitehead, Process and reality: An essay in cosmology, Macmillan Co., New York, 1929

[16] A. N. Whitehead, Science and the modern world, The Great Books of the Western World, 55, MacMillan Co., New Work, 1925