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SYMMETRY GEOMETRY BY PAIRINGS

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

Abstract

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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Keywords

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SYMMETRY GEOMETRY BY PAIRINGS

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

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