New rough approximations for n-cycles and n-paths

Sanahuja, Silvia Martínez

doi:10.1016/j.amc.2015.11.052

Cited by 15 publications

(2 citation statements)

References 21 publications

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“…In other terms, we may consider the table T (W) as a sort of numerical completion of M(W). The map γ W and some related averages have been broadly used in [16,33] in order to investigate the transmission of symmetry in some basic digraph and graph families.…”

Section: Introductionmentioning

confidence: 99%

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

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We call a quadruple $\mathcal{W}:=\langle F,U,\Omega,\Lambda \rangle$, where $U$ and $\Omega$ are two given non-empty finite sets, $\Lambda$ is a non-empty set and $F$ is a map having domain $U\times \Omega$ and codomain $\Lambda$, a pairing on $\Omega$. With this structure we associate a set operator $M_{\mathcal{W}}$ by means of which it is possible to define a preorder $\ge_{\mathcal{W}}$ on the power set $\mathcal{P}(\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\mathbb{L}$ there exist a finite set $\Omega_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $\Omega_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(\Omega_\mathbb{L}), \ge_\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\mathbb{L}$. In the second result, we prove that when the lattice $\mathbb{L}$ is endowed with an order-reversing involutory map $\psi: L \to L$ such that $\psi(\hat 0_{\mathbb{L}})=\hat 1_{\mathbb{L}}$, $\psi(\hat 1_{\mathbb{L}})=\hat 0_{\mathbb{L}}$, $\psi(\alpha) \wedge \alpha=\hat 0_{\mathbb{L}}$ and $\psi(\alpha) \vee \alpha=\hat 1_{\mathbb{L}}$, there exist a finite set $\Omega_{\mathbb{L},\psi}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\mathbb{L}$.

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Section: Introductionmentioning

confidence: 99%

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

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“…A new type of symmetry geometry has been recently developed for some set operators and set systems induced by graphs [17,18], digraphs [14,28], information tables [12], some types of discrete dynamical systems [11,13,16] and metric spaces [17]. This geometry is based on a notion of generalized symmetry between c 2018 Australian Mathematical Publishing Association Inc. subsets of points (for metric spaces) or vertices (for graphs and digraphs), and we refer the reader to [17,18] for details concerning these notions and corresponding results.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Geometry by Pairings

Chiaselotti

Gentile

Infusino

2018

J. Aust. Math. Soc.

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In this paper, we introduce asymmetry geometryfor 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. Apairing$\mathfrak{P}$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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