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Lattice Representations with Set Partitions Induced by Pairings
We call a quadruple , where and are two given non-empty finite sets, is a non-empty set and is a map having domain and codomain , a pairing on . With this structure we associate a set operator by means of which it is possible to define a preorder on the power set 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 there exist a finite set and a pairing on such that the quotient of the preordered set with respect to its symmetrization is a lattice that is order-isomorphic to . In the second result, we prove that when the lattice is endowed with an order-reversing involutory map such that , , and , there exist a finite set and a pairing on it inducing a specific poset which is order-isomorphic to