Dependency and accuracy measures for directed graphs

Chiaselotti, Giampiero; Gentile, Tommaso; Infusino, Federico; Oliverio, Paolo Antonio

doi:10.1016/j.amc.2017.10.031

Cited by 7 publications

(1 citation statement)

References 47 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%