The Consequences of Dilworth’s Work on Lattices with Unique Irreducible Decompositions

Abstract

In 1937, G. Birkhoff [6] proved that every element of a finite dimensional distributive lattice L has a “unique irredundant decomposition” as meet of meetirreducible elements (or as a join of join-irreducible elements). What does this mean? Let us denote by M(L) or simply M (resp. J(L) or J) the set of all meetirreducible (resp. join-irreducible) elements of a lattice L; then for each element x of a finite dimensional distributive lattice there exists a unique subset of M such that x = ∧S and x < ∧(S − {m) for every m in S (and dually for J). It is easy to find non-distributive lattices having this property of unicity of irredundant meet decomposition (the simplest example is given in Figure 1 of the Background for this chapter). Thus a natural question was to characterize such (finite dimensional) lattices; that is exactly what the Dilworth did in [17]. I don’t know how [17] was received in 1940 (I was two years old...), but it seems it was considered as an ending: a natural question was nicely solved, so there was nothing more to say. And indeed nothing more was said on these lattices for twenty years. But in mathematics fire can spring from coals: since 1960, at least fifty papers have been published that directly or indirectly concern or rediscover the lattices characterized in [17] (or their duals); moreover these lattices appear at the core of fast-developing combinatorial theories, like convex geometries (Edelman and Jamison [28]), greedoids (Korte and Lovasz [36]), or exchange systems (Brylawski and Dieter [11]).