8 - Local Modularity

Summary

The Kurosh–Ore Replacement Property

Summary. The Kurosh–Ore theorem for finite decompositions in modular lattices was given in Section 1.6. In particular we have seen that modularity implies the Kurosh–Ore replacement property for meet decompositions (∧-KORP). On the other hand, the lattice N₅ shows that the ∧-KORP (together with its dual, the ∨-KORP) does not even imply semimodularity. Thus the question arose how to characterize the Kurosh–Ore replacement property in general and, in particular, in the semimodular case. The pertinent fundamental results are due to Dilworth and Crawley. In this section we have a brief look at the characterization of the ∧-KORP for strongly atomic algebraic lattices. In the following section we turn to the semimodular case.

The equivalence of the ∧-KORP, Crawley's condition (Cr*), and dual consistency for lattices of finite length was mentioned in Theorem 4.5.1.

From Section 1.8 we recall the definition of completely meet-irreducible elements and the fact that in an algebraic lattice every element is a meet of completely meet-irreducible elements, that is, infinite meet decompositions exist (cf. Theorem 1.8.1). We also recall that if an algebraic lattice is strongly atomic, then any meet-irreducible element is completely meet-irreducible, that is, the two concepts are identical. Moreover, Crawley [1961] proved the existence of irredundant meet decompositions (cf. Theorem 1.8.2).

In Section 1.8 the ∧-KORP was defined for complete lattices. Crawley's condition (Cr*) (defined in Section 4.5) can also be formulated for strongly atomic algebraic lattices. Finally let us state that we may define consistency for strongly dually atomic dually algebraic lattices as Kung [1985] did for lattices of finite length (cf. Section 4.5). In a dual way we define dual consistency for strongly atomic algebraic lattices.