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  • G. CHIASELOTTI (a1), T. GENTILE (a2) and F. INFUSINO (a3)


In this paper, we introduce a symmetry geometry for all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let $\unicode[STIX]{x1D6FA}$ be a given set. A pairing $\mathfrak{P}$ on $\unicode[STIX]{x1D6FA}$ is a triple $\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$ , where $U$ and $\unicode[STIX]{x1D6EC}$ are nonempty sets and $F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$ is a map having domain $U\times \unicode[STIX]{x1D6FA}$ and codomain $\unicode[STIX]{x1D6EC}$ . Through this notion, we introduce a local symmetry relation on $U$ and a global symmetry relation on the power set ${\mathcal{P}}(\unicode[STIX]{x1D6FA})$ . Based on these two relations, we establish the basic properties of our symmetry geometry induced by $\mathfrak{P}$ . The basic tool of our study is a closure operator $M_{\mathfrak{P}}$ , by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.


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[1] Alspach, B., ‘Isomorphism and Cayley graphs on abelian groups’, in: Graph Symmetry (eds. Hahn, G. and Sabidussi, G.) (Kluwer, Dordrecht, 1997), 122.
[2] Alspach, B., Heinrich, K. and Varma, B. M., ‘Decompositions of complete symmetric digraphs into the oriented pentagons’, J. Aust. Math. Soc. A 28 (1979), 353361.
[3] Alspach, B., Marusic, D. and Nowitz, L., ‘Constructing graphs which are 1/2-transitive’, J. Aust. Math. Soc. A 56 (1994), 391402.
[4] Batten, L. M., ‘Rank functions of closure spaces of finite rank’, Discrete Math. 49(2) (1984), 113116.
[5] Batten, L. M. and Beutelspacher, A., The Theory of Finite Linear Spaces-Combinatorics of Points and Lines (Cambridge University Press, New York, 1993).
[6] Batten, L. M. and Dover, J. M., ‘Blocking semiovals of type (1, m + 1, n + 1)’, SIAM J. Discrete Math. 14(4) (2001), 446457.
[7] Bayley, R. A., ‘Orthogonal partitions in designed experiments’, Des. Codes Cryptogr. 8(3) (1996), 4577.
[8] Bayley, R. A., Association Schemes: Designed Experiments, Algebra and Combinatorics (Cambridge University Press, Cambridge, 2004).
[9] Berge, C., Hypergraphs: Combinatorics of Finite Sets (Elsevier, Amsterdam, 1984).
[10] Birkhoff, G., Lattice Theory, 3rd edn (American Mathematical Society, Providence, RI, 1967).
[11] Bisi, C. and Chiaselotti, G., ‘A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture’, Adv. Geom. 13(1) (2013), 127.
[12] Bisi, C., Chiaselotti, G., Ciucci, D., Gentile, T. and Infusino, F., ‘Micro and macro models of granular computing induced by the indiscernibility relation’, Inf. Sci. 388–389 (2017), 247273.
[13] Bisi, C., Chiaselotti, G., Gentile, T. and Oliverio, P. A., ‘Dominance order on signed partitions’, Adv. Geom. 17(1) (2017), 529.
[14] Bisi, C., Chiaselotti, G., Marino, G. and Oliverio, P. A., ‘A natural extension of the Young partition lattice’, Adv. Geom. 15(3) (2015), 263280.
[15] Bonacini, P., Gionfriddo, M. and Marino, L., ‘Nesting House-designs’, Discrete Math. 339(4) (2016), 12911299.
[16] Cattaneo, G., Chiaselotti, G., Oliverio, P. A. and Stumbo, F., ‘A new discrete dynamical system of signed integer partitions’, European J. Combin. 55 (2016), 119143.
[17] Chiaselotti, G., Gentile, T. and Infusino, F., ‘Simplicial complexes and closure systems induced by indistinguishability relations’, C. R. Acad. Sci. Paris, Ser. I 355 (2017), 9911021.
[18] Chiaselotti, G., Gentile, T., Infusino, F. and Oliverio, P. A., ‘The adjacency matrix of a graph as a data table. A geometric perspective’, Ann. Mat. Pura Appl. (4) 196(3) (2017), 10731112.
[19] Dohmen, K., ‘A broken-circuits-theorem for hypergraphs’, Arch. Math. 64 (1995), 159162.
[20] Eiter, T. and Gottlob, G., ‘Identifying the minimal transversals of a hypergraph and related problems’, SIAM J. Comput. 24 (1995), 12781304.
[21] Kelarev, A., Ryan, J. and Yearwood, J., ‘Cayley graphs as classifiers for data mining: the influence of asymmetries’, Discrete Math. 309 (2009), 53605369.
[22] Ksontini, R., ‘The fundamental group of the Quillen complex of the symmetric group’, J. Algebra 282 (2004), 3357.
[23] Kunzi, H.-P. A. and Yildiz, F., ‘Convexity structures in T 0 -quasi-metric spaces’, Topology Appl. 200 (2016), 218.
[24] Lee, S. and Shih, M., ‘Sperner matroid’, Arch. Math. 81 (2003), 103112.
[25] Magner, A., Janson, S., Kollias, G. and Szpankowski, W., ‘On symmetry of uniform and preferential attachment graphs’, Electron. J. Combin. 21(3) (2014).
[26] Martin, H. W., ‘Metrization of symmetric spaces and regular maps’, Proc. Amer. Math. Soc. 35 (1972), 269274.
[27] Rieder, J., ‘The lattices of matroid bases and exact matroid bases’, Arch. Math. 56 (1991), 616623.
[28] Sanahuja, S. M., ‘New rough approximations for n-cycles and n-paths’, Appl. Math. Comput. 276 (2016), 96108.
[29] Welsh, D. J. A., Matroid Theory (Academic Press, New York, 1976).
[30] Xiang, S. W. and Yang, H., ‘Some properties of abstract convexity structures on topological spaces’, Nonlinear Anal. 67 (2007), 803808.
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  • G. CHIASELOTTI (a1), T. GENTILE (a2) and F. INFUSINO (a3)


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