Alexandroff topologies and monoid actions

  • 1 Department of Mathematics and Computer Science, University of Calabria, Via Pietro Bucci, Cubo 30B, 87036, Arcavacata di Rende, Italy
  • 2 Department of Mathematics and Computer Science, University of Calabria, Via Pietro Bucci, Cubo 30B, 87036, Arcavacata di Rende, Italy
Giampiero ChiaselottiORCID iD: https://orcid.org/0000-0003-2247-2386 and Federico G. Infusino
  • Department of Mathematics and Computer Science, University of Calabria, Via Pietro Bucci, Cubo 30B, 87036, Arcavacata di Rende, (CS), Italy
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

Given a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X. Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family 𝒜σ,X of the closed subsets Y of X such that, for any y1,y2Y, there exists a third element yY whose closure contains both y1 and y2. More in detail, relying on some specific properties of the maximal members of the family 𝒜σ,X, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

  • [1]

    J. A. Aledo, L. G. Diaz, S. Martinez and J. C. Valverde, Predecessors and Garden-of-Eden configurations in parallel dynamical systems on maxterm and minterm Boolean functions, J. Comput. Appl. Math. 348 (2019), 26–33.

    • Crossref
    • Export Citation
  • [2]

    J. A. Aledo, L. G. Diaz, S. Martinez and J. C. Valverde, Solution to the predecessors and Gardens-of-Eden problems for synchronous systems over directed graphs, Appl. Math. Comput. 347 (2019), 22–28.

  • [3]

    P. Alexandroff, Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501–518.

  • [4]

    S. J. Andima and W. J. Thron, Order-induced topological properties, Pacific J. Math. 75 (1978), no. 2, 297–318.

    • Crossref
    • Export Citation
  • [5]

    A. Bailey, M. Finn-Sell and R. Snocken, Subsemigroup, ideal and congruence growth of free semigroups, Israel J. Math. 215 (2016), no. 1, 459–501.

    • Crossref
    • Export Citation
  • [6]

    A. Bailey and J. H. Renshaw, Covers of acts over monoids and pure epimorphisms, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 3, 589–617.

    • Crossref
    • Export Citation
  • [7]

    C. Bisi, On commuting polynomial automorphisms of k \mathbb{C}^{k}, k 3 k\geq 3, Math. Z. 258 (2008), no. 4, 875–891.

    • Crossref
    • Export Citation
  • [8]

    C. Bisi, On closed invariant sets in local dynamics, J. Math. Anal. Appl. 350 (2009), no. 1, 327–332.

    • Crossref
    • Export Citation
  • [9]

    C. Bisi, A Landau’s theorem in several complex variables, Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 737–742.

    • Crossref
    • Export Citation
  • [10]

    P. Bonacini, M. Gionfriddo and L. Marino, Nesting house-designs, Discrete Math. 339 (2016), no. 4, 1291–1299.

    • Crossref
    • Export Citation
  • [11]

    G. Chiaselotti, T. Gentile and F. Infusino, Simplicial complexes and closure systems induced by indistinguishability relations, C. R. Math. Acad. Sci. Paris 355 (2017), no. 9, 991–1021.

    • Crossref
    • Export Citation
  • [12]

    G. Chiaselotti, T. Gentile and F. Infusino, Decision systems in rough set theory: A set operatorial perspective, J. Algebra Appl. 18 (2019), no. 1, Article ID 1950004.

  • [13]

    G. Chiaselotti, T. Gentile and F. Infusino, Local dissymmetry on graphs and related algebraic structures, Internat. J. Algebra Comput. 29 (2019), no. 8, 1499–1526.

    • Crossref
    • Export Citation
  • [14]

    G. Chiaselotti, T. Gentile and F. Infusino, New perspectives of granular computing in relation geometry induced by pairings, Ann. Univ. Ferrara Sez. VII Sci. Mat. 65 (2019), no. 1, 57–94.

    • Crossref
    • Export Citation
  • [15]

    G. Chiaselotti, T. Gentile, F. G. Infusino and P. A. Oliverio, The adjacency matrix of a graph as a data table: A geometric perspective, Ann. Mat. Pura Appl. (4) 196 (2017), no. 3, 1073–1112.

    • Crossref
    • Export Citation
  • [16]

    G. Chiaselotti, F. Infusino and P. A. Oliverio, Set relations and set systems induced by some families of integral domains, Adv. Math. 363 (2020), Article ID 106999.

  • [17]

    B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder, European J. Combin. 44 (2015), 208–217.

    • Crossref
    • Export Citation
  • [18]

    B. Davvaz and M. Karimian, On the γ n \gamma^{\ast}_{n}-complete hypergroups, European J. Combin. 28 (2007), no. 1, 86–93.

    • Crossref
    • Export Citation
  • [19]

    M. A. Erdal and O. Ünlü, Semigroup actions on sets and the Burnside ring, Appl. Categ. Structures 26 (2018), no. 1, 7–28.

    • Crossref
    • Export Citation
  • [20]

    M. Gionfriddo, E. Guardo and L. Milazzo, Extending bicolorings for Steiner triple systems, Appl. Anal. Discrete Math. 7 (2013), no. 2, 225–234.

    • Crossref
    • Export Citation
  • [21]

    J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, New Math. Monogr. 22, Cambridge University, Cambridge, 2013.

  • [22]

    D. Hofmann, Topological theories and closed objects, Adv. Math. 215 (2007), no. 2, 789–824.

    • Crossref
    • Export Citation
  • [23]

    D. Hofmann, G. J. Seal and W. Tholen (eds.), Monoidal Topology, Encyclopedia Math. Appl. 153, Cambridge University, Cambridge, 2014.

  • [24]

    P. T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3, Cambridge University, Cambridge, 1986.

  • [25]

    J. D. Lawson, Points of continuity for semigroup actions, Trans. Amer. Math. Soc. 284 (1984), no. 1, 183–202.

    • Crossref
    • Export Citation
  • [26]

    S. Lazaar, T. Richmond and H. Sabri, Homogeneous functionally Alexandroff spaces, Bull. Aust. Math. Soc. 97 (2018), no. 2, 331–339.

    • Crossref
    • Export Citation
  • [27]

    S. Lazaar, T. Richmond and H. Sabri, The autohomeomorphism group of connected homogeneous functionally Alexandroff spaces, Comm. Algebra 47 (2019), no. 9, 3818–3829.

    • Crossref
    • Export Citation
  • [28]

    M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465–474.

  • [29]

    G. J. Seal, Order-adjoint monads and injective objects, J. Pure Appl. Algebra 214 (2010), no. 6, 778–796.

    • Crossref
    • Export Citation
  • [30]

    A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379–398.

    • Crossref
    • Export Citation
  • [31]

    W. Tholen, A categorical guide to separation, compactness and perfectness, Homology Homotopy Appl. 1 (1999), 147–161.

    • Crossref
    • Export Citation
  • [32]

    A. Weston, On the generalized roundness of finite metric spaces, J. Math. Anal. Appl. 192 (1995), no. 2, 323–334.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

Search