Lattices with Unique Irreducible Decompositions
R. P. Dilworth · Birkhäuser Boston eBooks · 1990
Consider a lattice S in which the ascending chain condition holds. Then each element of S has at least one reduced 1 representation as a cross-cut of irreducibles. Now it is well known that the requirement that this representation be unique considerably restricts the structure of the lattice. For example, Garrett Birkhoff [1] has proved that a modular lattice in which every element is uniquely expressible as a reduced cross-cut of irreducibles is distributive. Furthermore, Morgan Ward has shown that unicity of the irreducible decompositions implies that the lattice is a Birkhoff lattice. 2 These results suggest the interesting problem of characterizing a lattice in which every element has a unique reduced representation as a cross-cut of irreducibles in terms of the structure of the lattice. We give here a complete solution of this problem. We show, namely, that such lattices are simply those Birkhoff lattices in which every modular sublattice is distributive.