Lattice Representations with Set Partitions Induced by Pairings

Abstract

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 $\mathbb{L}$.

Real subset sums and posets with an involution

“…Involution partially ordered sets (briefly iinvolution posets) are order structures classically used to describe specific combinatorial aspects in Ockam algebras and de Morgan algebras (see [9] for more details). These posets generalize the classical Boolean algebras [23,38], and in literature they are studied in several different mathematical contexts (see for example [1,11,16,19,25,26,30,31,36,37]). In this paper we use involution posets in a set operatorial perspective, in analogy with what has been recently done for integral domains [14] and monoid actions [15].…”