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ResumenLa matemática difusa generaliza los conceptos tradicionales de la matemática utilizando los llamados conjuntos difusos, lo que permite modelar y estudiar de manera más apropiada fenómenos caracterizados por su imprecisión. Estas generalizaciones incluyen conceptos de álgebra, análisis y topología, entre otros. Por otra parte, los complejos cúbicos tienen aplicaciones al procesamiento de imágenes digitales y al estudio de los sistemas dinámicos, pero en la literatura actual no hay una extensión de sus propiedades utilizando conjuntos difusos. En este documento se propone una generalización del concepto de complejo cúbico y de algunas de sus características, tales como realización poliédrica, conexidad, componente conexa y agujero, utilizando conjuntos difusos. Se definen los árboles superior e inferior de un complejo cúbico difuso, los cuales proporcionan información sobre la manera en la que están relacionados sus extremos regionales. También se definen los grupos de homología de estos complejos cúbicos difusos y se demuestra que el rango del 0-grupo de homología de un nivel dado es igual al número de máximos regionales de dicho nivel. Por último, se muestra cómo se le puede asociar un complejo cúbico difuso a una imagen digital bidimensional en tonos de gris para estudiar algunas de sus propiedades topológicas. AbstractFuzzy mathematics generalize concepts of traditional mathematics using fuzzy sets. This enables to study and model more properly phenomens characterized by imprecision. These generalizations includes concepts of algebra, analysis and topology. On the other side, cubical complexes have applications in digital image processing and in the study of dynamical systems, but in the actual literature there is not an extension of their properties using fuzzy sets. In this paper is proposed a generalization of the concept of cubical complex and of some of their properties, such as connectedness, polyhedral realization, connected component and holes, using fuzzy sets. The upper and lower trees of a fuzzy cubical complex are defined, which give information about the way in which its regional extrema are related. The homology groups of a fuzzy cubical complex are defined and it is shown that the rank of the 0-homology group of a given level is equal with the number of regional maxima of that level. Finally, it is shown how to associate a fuzzy cubical complex with a bidimensional digital grayscale image in order to study somo of its topological properties.

A binary three-dimensional (3D) image II is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I)Q(I) (embedded in R3R3) associated to II to obtain a polyhedral complex P(I)P(I) homotopy equivalent to Q(I)Q(I) such that the boundary surface of P(I)P(I) is a 2D manifold (and, hence, P(I)P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex KK, called ExtendedCubeMap (ECM) representation of KK, that codifies: (1) the information of the cells of KK (including geometric information), under the form of a 3D grayscale image gPgP; and (2) the boundary face relations between the cells of KK, under the form of a set BPBP of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation EQEQ of the cubical complex Q(I)Q(I) to obtain an ECM representation of a well-composed polyhedral complex P(I)P(I) that is homotopy equivalent to Q(I)Q(I). The construction of the polyhedral complex P(I)P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image gPgP (obtained by the repairing process on EQEQ) together with the set BPBP codify all the geometric and topological information of P(I)P(I).

A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q(I ), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59–77, 2015), we presented a method to “locally repair” Q(I ) to obtain a polyhedral complex P(I ) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q(I ), satisfying that its boundary surface is a 2D manifold. P(I ) is called a well-composed polyhedral complex over the picture I . Besides, we developed a new codification system for P(I ), encoding geometric information of the cells of P(I ) under the form of a 3D grayscale image, and the boundary face relations of the cells of P(I ) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P(I ), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this “minimal” codification of P(I ), we finally present a method to compute the 2-cells in the boundary surface of P(I ). Ministerio de Economía y Competitividad MTM2015-67072-P

We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical cells. An implementation in C++ of the introduced algorithms and some examples are available at http://www.pawelpilarczyk.com/chaincon/. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it., This research was partially supported from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE - Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundacao para a Ciencia e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008), as well as from the funds distributed through the European Science Foundation (ESF) Research Networking Programme on "Applied and Computational Algebraic Topology" (ACAT). P. Real was additionally supported by the Spanish Ministry of Science and Innovation, project no. MTM2009-12716.

The geometric models of higher dimensional automata (HDA) and Dijkstra's PV-model are cubically subdivided topological spaces with a local partial order. If a cubicalization of a topological space is free of immersed cubic Möbius bands, then there are consistent choices of direction in all cubes, such that any n-cube in the cubic subdivision is dihomeomorphic to [0,1]n with the induced partial order from Rn. After subdivision once, any cubicalized space has a cubical local partial order. In particular, all triangularized spaces have a cubical local partial order. This implies in particular that the underlying geometry of an HDA may be quite complicated.

We study finite foldable cubical complexes of nonpositive curvature (in the sense of A.D. Alexandrov). We show that such a complex X admits a graph of spaces decomposition. It is also shown that when dim X=3, X contains a closed rank one geodesic in the 1-skeleton unless the universal cover of X is isometric to the product of two CAT(0) cubical complexes., Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-28.abs.html

Any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satisfies the Baum–Connes conjecture. Any diagram group over a finite presentation of a finite semigroup is of type F ∞ .

The cohomology of digraphs was introduced for the first time by Dimakis and Muller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph $G_B$ of the partially ordered set $B_{S}$ of simplexes of a simplicial complex $S$ has graph homology that is isomorphic to the simplicial homology of $S$. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph $G_{S}$ of $G_B$, whose homologies are isomorphic to the simplicial homologies of $S$.

We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin's enumeration of spanning trees of a complete colorful simplicial complex from the cellular Matrix-Tree Theorem together with a result of Kook, Reiner and Stanton., 24 pages, revised version, to appear in Advances in Applied Mathematics

In the geometric realization of a cubical complex without degeneracies, a $\Box$-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipahts are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting $\Box$-set. And that two combinatorial dipaths which are dihomotopic threough a non-combinatorial dihomogopy are in fact combinatorially dihomotopic, in a geometric $\Box$-set. Moreover, we prove that in a geometric $\Box$-set, the d-homotopy introdced in [M. Grandis (2003)] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen (1999)]. Udgivelsesdato: AUG In the geometric realization of a cubical complex without degeneracies, a $\Box$-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipahts are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting $\Box$-set. And that two combinatorial dipaths which are dihomotopic threough a non-combinatorial dihomogopy are in fact combinatorially dihomotopic, in a geometric $\Box$-set. Moreover, we prove that in a geometric $\Box$-set, the d-homotopy introdced in [M. Grandis (2003)] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen (1999)].

We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension., Comment: 27 pages, 7 figures