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 M_W 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 Ω_L and a pairing W on Ω_L such that the quotient of the preordered set (P(Ω_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 ψ: L → L such that ψ(0_L )=1_L , ψ(1_L )=0_L , ψ(α) α=0_L and ψ(α) α=1_L , there exist a finite set Ω_L,ψ and a pairing on it inducing a specific poset which is order-isomorphic to L. , number=1, journal=The Electronic Journal of Combinatorics, author=Chiaselotti, Giampiero and Gentile, Tommaso and Infusino, Federico, year=2020, month=Jan., pages=#P1.19