Lattice representations with set partitions induced by pairings

We call a quadruple W := ⟨F, U, Ω, Λ⟩, 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 MW by means of which it is possible to define a preorder W on the power set 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 L there exist a finite set ΩL and a pairing W on ΩL such that the quotient of the preordered set (P(ΩL),W) with respect to its symmetrization is a lattice that is order-isomorphic to L. In the second result, we prove that when the lattice L is endowed with an order-reversing involutory map ψ : L → L such that ψ(ˆ0L) = ˆ1L, ψ(ˆ1L) = ˆ0L, ψ(α)∧α = ˆ0L and ψ(α) ∨ α = ˆ1L, there exist a finite set ΩL,ψ and a pairing on it inducing a specific poset which is order-isomorphic to L.