Lattice Representations with Set Partitions Induced by Pairings - Chiaselotti, Giampiero; Gentile, Tommaso; Infusino, Federico (The Electronic Journal of Combinatorics, 2020)
We call a quadruple $\mathcal{W}:=\langle F,U,Ω,Λ \rangle$, where $U$ and $Ω$ are two given non-empty finite sets, $Λ$ is a non-empty set and $F$ is a map having domain $U× Ω$ and codomain $Λ$, a pairing on $Ω$. 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 $\mathcal{P}(Ω)$ 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 $Ω_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $Ω_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(Ω_\mathbb{L}), \ge_\mathcal{W})$ 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 $ψ: L → L$ such that $ψ(\hat 0_{\mathbb{L}})=\hat 1_{\mathbb{L}}$, $ψ(\hat 1_{\mathbb{L}})=\hat 0_{\mathbb{L}}$, $ψ(α) \wedge α=\hat 0_{\mathbb{L}}$ and $ψ(α) \vee α=\hat 1_{\mathbb{L}}$, there exist a finite set $Ω_{\mathbb{L},ψ}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\mathbb{L}$.