Any distributive lattice is determined by a semigroup from a certain class (Q1068869)

Let \(R=(R,\vee,\wedge)\) be a distributive lattice. Define for any (a,b), (c,d)\(\in R\times R\) \((a,b)\circ (c,d)=(a\vee (b\wedge c),b\wedge d)\). Then \(S(R)=(R\times R,\circ)\) becomes a semigroup. It is easy to show that S(R) is an idempotent semigroup in which the Green relation \({\mathcal R}\) is trivial: a\({\mathcal R}b\Leftrightarrow a=b\). Hence \({\mathcal J}={\mathcal L}\) on S(R). Moreover \({\mathcal J}\) is a congruence on S(R), and S(R) is a semilattice of left zero semigroups. The aim of the paper is to prove the following: Theorem. Let \(R=(R,\vee,\wedge)\) and \(F=(F,\vee,\wedge)\) be two distributive lattices. Then the lattices R and F are isomorphic if and only if the semigroups S(R) and S(F) are isomorphic.