## Alexandroff topologies and monoid actions.(English)Zbl 1440.54001

In this paper, the statement “Any Alexandroff topology may be obtained through a monoid action” is provided. Several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions are introduced. A specific notion of dependence based on the union of subsets is introduced for an Alexandroff space $$X$$ with associated topological closure operator $$\sigma$$. The authors study the family $$\mathcal{A}_{\sigma ,X}$$ of closed subsets $$Y$$ of $$X$$ such that, for any $$y_{1}$$, $$y_{2}\in Y$$, there exists a third element $$y\in Y$$ whose closure contains both $$y_{1}$$ and $$y_{2}$$. A decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence is provided.

### MSC:

 54A05 Topological spaces and generalizations (closure spaces, etc.) 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 20M30 Representation of semigroups; actions of semigroups on sets 20M15 Mappings of semigroups

### Keywords:

Alexandroff spaces; closure operators; monoids; monoid actions
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### References:

 [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. · Zbl 1404.37043 [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. · Zbl 1428.37042 [3] P. Alexandroff, Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501-518. · Zbl 0018.09105 [4] S. J. Andima and W. J. Thron, Order-induced topological properties, Pacific J. Math. 75 (1978), no. 2, 297-318. · Zbl 0384.54018 [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. · Zbl 1398.20066 [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. · Zbl 1309.20053 [7] C. Bisi, On commuting polynomial automorphisms of \mathbb{C}^k, k\geq 3, Math. Z. 258 (2008), no. 4, 875-891. · Zbl 1161.32006 [8] C. Bisi, On closed invariant sets in local dynamics, J. Math. Anal. Appl. 350 (2009), no. 1, 327-332. · Zbl 1151.37315 [9] C. Bisi, A Landau’s theorem in several complex variables, Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 737-742. · Zbl 1366.32008 [10] P. Bonacini, M. Gionfriddo and L. Marino, Nesting house-designs, Discrete Math. 339 (2016), no. 4, 1291-1299. · Zbl 1329.05241 [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. · Zbl 1371.05327 [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. · Zbl 1411.68155 [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. · Zbl 1423.05079 [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. · Zbl 07074946 [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. · Zbl 1366.05029 [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. · Zbl 1441.13025 [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. · Zbl 1308.06010 [18] B. Davvaz and M. Karimian, On the \gamma^{\ast}_n-complete hypergroups, European J. Combin. 28 (2007), no. 1, 86-93. · Zbl 1117.20053 [19] M. A. Erdal and O. Ünlü, Semigroup actions on sets and the Burnside ring, Appl. Categ. Structures 26 (2018), no. 1, 7-28. · Zbl 1409.16033 [20] M. Gionfriddo, E. Guardo and L. Milazzo, Extending bicolorings for Steiner triple systems, Appl. Anal. Discrete Math. 7 (2013), no. 2, 225-234. · Zbl 1408.05096 [21] J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, New Math. Monogr. 22, Cambridge University, Cambridge, 2013. · Zbl 1280.54002 [22] D. Hofmann, Topological theories and closed objects, Adv. Math. 215 (2007), no. 2, 789-824. · Zbl 1127.18001 [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. · Zbl 0516.54031 [26] S. Lazaar, T. Richmond and H. Sabri, Homogeneous functionally Alexandroff spaces, Bull. Aust. Math. Soc. 97 (2018), no. 2, 331-339. · Zbl 1395.54016 [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. · Zbl 07076205 [28] M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465-474. · Zbl 0142.21503 [29] G. J. Seal, Order-adjoint monads and injective objects, J. Pure Appl. Algebra 214 (2010), no. 6, 778-796. · Zbl 1203.18009 [30] A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379-398. · Zbl 0139.15905 [31] W. Tholen, A categorical guide to separation, compactness and perfectness, Homology Homotopy Appl. 1 (1999), 147-161. · Zbl 0924.18003 [32] A. Weston, On the generalized roundness of finite metric spaces, J. Math. Anal. Appl. 192 (1995), no. 2, 323-334. · Zbl 0842.54034
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