Oriented components and their separations

• Autores: Baerbel M R Stadler, Peter F. Stadler
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 18, Nº. 2, 2017, págs. 255-275
• Idioma: inglés
• DOI: 10.4995/agt.2017.5868
• Enlaces
  • Texto completo
• Resumen

  • There is a tight connection between connectedness, connected components, and certain types of separation spaces. Recently, axiom systems for oriented connectedness were proposed leading to the notion of reaches. Here, we introduce production relations as a further generalization of connectivity spaces and reaches and derive associated systems of oriented components that generalize connected components in a natural manner. The main result is a characterization of generalized reaches in terms of equivalent separation spaces.

• Referencias bibliográficas
  • J. L. Andersen, C. Flamm, D. Merkle and P. F. Stadler, Generic strategies for chemical space exploration, Int. J. Comp. Biol. Drug Design...
  • G. Benkö, F. Centler, P. Dittrich, C. Flamm, B. M. R. Stadler and P. F. Stadler, A topological approach to chemical organizations, Alife 15...
  • G. Berry, The chemical abstract machine, Theor. Comp. Sci. 96 (1992), 217-248. https://doi.org/10.1016/0304-3975(92)90185-I
  • R. Börger, Connectivity spaces and component categories, in: H. L. Bentley, H. Herrlich, M. Rajagopalan, H. Wolff (Eds.), Categorical topology,...
  • M. Changat and J. Mathew, Induced path transit function, monotone and {Peano} axioms, Discrete Math. 286 (2004), 185-194. https://doi.org/10.1016/j.disc.2004.02.017
  • S. Dugowson, Les frontières dialectiques, Math. Social Sci. 177 (2007), 87-152. https://doi.org/10.4000/msh.3908
  • M. Erné and R. Vainio, Connectivity in lattice-ordered spaces, Math. Nachr. 147 (1990), 13-28. https://doi.org/10.1002/mana.19901470103
  • L. G. Fearnley, M. J. Davis, M. A. Ragan and L. K. Nielsen, Extracting reaction networks from databases - opening Pandora's box, Brief...
  • C. Flamm, B. M. R. Stadler and P. F. Stadler, Saddles and barrier in landscapes of generalized search operators, in: C. R. Stephens, M. Toussaint,...
  • C. Flamm, B. M. R. Stadler and P. F. Stadler, Generalized topologies: hypergraphs, chemical reactions, and biological evolution, in: S. C....
  • J. H. Holland, Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, MI, 1975.
  • K. Klemm, J. Qin and P. F. Stadler, Geometry and coarse-grained representations of landscapes, in: A. Engelbrecht, H. Richter (Eds.), Recent...
  • S. Leader, Local proximity spaces, Math. Annalen 169 (1957), 275-281. https://doi.org/10.1007/BF01362349
  • M. W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. 15 (1964), 417-422. https://doi.org/10.1090/S0002-9939-1964-0161305-3
  • L. Mazo, N. Passat, M. Couprie and C. Ronse, Digital imaging: a unified topological framework, J. Math. Imaging Vision 44 (2012), 19-37. https://doi.org/10.1007/s10851-011-0308-9
  • H. M. Mulder, The Interval Function of a Graph, Vol. 132 of Math. Centre Tracts, Math. Centre, Amsterdam, NL, 1980.
  • J. Muscat and D. Buhagiar, Connective spaces, Mem. Fac. Sci. Eng. Shimane Univ. Series B: Math. Sci. 39 (2006), 1-13.
  • W. J. Pervin, On separation and proximity spaces, Amer. Math. Monthly 71 (1964), 158-161. https://doi.org/10.2307/2311745
  • I. Rechenberg, Evolutionstrategie: Optimierung Technischer Systeme nach Prinzipien des Biologischen Evolution, Frommann-Holzboog Verlag, Stuttgart,...
  • C. Ronse, Set-theoretical algebraic approaches to connectivity in continuous or digital spaces, J. Math. Imaging Vision 8 (1998), 41-58. https://doi.org/10.1023/A:1008210216583
  • C. Ronse, Partial partitions, partial connections and connective segmentation, J. Math. Imaging Vis. 32 (2008), 97-125. https://doi.org/10.1007/s10851-008-0090-5
  • C. Ronse, Axiomatics for oriented connectivity, Pattern Recognition Lett. 47 (2014), 120-128. https://doi.org/10.1016/j.patrec.2014.03.020
  • J. Serra, Mathematical morphology for boolean lattices, in: J. Serra (Ed.), Image Analysis and Mathematical Morphology, Theoretical Advances,...
  • J. M. Smirnov, On proximity spaces, Mat. Sb. 31 (1952), 543-574.
  • B. M. R. Stadler and P. F. Stadler, Generalized topological spaces in evolutionary theory and combinatorial chemistry, J. Chem. Inf. Comput....
  • B. M. R. Stadler and P. F. Stadler, Connectivity spaces, Math. Comp. Sci 9 (2015), 409-436. https://doi.org/10.1007/s11786-015-0241-1
  • B. M. R. Stadler, P. F. Stadler, G. Wagner and W. Fontana, The topology of the possible: Formal spaces underlying patterns of evolutionary...
  • O. Tankyevych, H. Talbot, N. Passat, Semi-connections and hierarchies, in: C. L. Luengo Hendriks, G. Borgefors, R. Strand (Eds.), Mathematical...
  • A. D. Wallace, Separation spaces, Ann. Math. 42 (1941), 687-697. https://doi.org/10.2307/1969257
  • S. Wright, The roles of mutation, inbreeding, crossbreeeding and selection in evolution, in: D. F. Jones (Ed.), Proceedings of the Sixth International...