THE UNIQUENESS OF THE DECOMPOSITION OF DISTRIBUTIVE LATTICES INTO SUMS OF BOOLEAN LATTICES.
Joanna Grygiel, Piotr Wojtylak · Reports on Mathematical Logic · 1997
b s t r a c t We prove that any decomposition of any nite distributive lattice into (the Wro nski sum of) Boolean algebras contains all maximal Boolean fragments of the lattice. The maximal elements of the decomposition are thus uniquely determined. We also exhibit a practical method of nding them. Let = and B = be lattices such that B is a lter in and an ideal in B, and the orderings and B coincide on B. Then [ B [ ( B), is a lattice ordering on [ B and the resulting lattice, called a sum of and B, is denoted by B. The sum operation was introduced by Wro nski [5], and its special case with A B = f1Ag = f0Bg by Troelstra [4]. In particular, if B is a two{element Boolean algebra then B is the same as , where is the Ja skowski operation of adding to the top element ( so called \mast ), see [2]. Kotas, Wojtylak [3] proved that the closure of the class of all nite