Lattice Representations with Set Partitions Induced by Pairings

We call a quadruple W := < F, U, Omega, A >, where U and Omega are two given non-empty finite sets, A is a non-empty set and F is a map having domain U x Omega and codomain A, a pairing on Omega. With this structure we associate a set operator M-w by means of which it is possible to define a preorder >=(w) on the power set 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 L there exist a finite set Omega(L) and a pairing W on Omega(L) such that the quotient of the preordered set (P(Omega(L)), >=(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 psi : L -> L such that psi((0) over cap (L)) = (1) over cap (L), psi((1) over cap (L)) = (0) over cap (L), psi(alpha) = boolean AND alpha = (0) over cap (L) and psi(alpha) boolean OR alpha = (1) over cap (L), there exist a finite set Omega(L,psi) and a pairing on it inducing a specific poset which is order-isomorphic to L.