# Kurosh-Ore theorem

###### Theorem 1 (Kurosh-Ore).

Let $L$ be a modular lattice and suppose that $a\in L$ has two irredundant decompositions of joins of join-irreducible elements:

 $a=x_{1}\vee\cdots\vee x_{m}=y_{1}\vee\cdots\vee y_{n}.$

Then

1. 1.

$m=n$, and

2. 2.

every $x_{i}$ can be replaced by some $y_{j}$, so that

 $a=x_{1}\vee\cdots\vee x_{i-1}\vee y_{j}\vee x_{i+1}\vee\cdots\vee x_{m}.$

There is also a dual statement of the above theorem in terms of meets.

Remark. Additionally, if $L$ is a distributive lattice, then the second property above (known the replacement property) can be strengthened: each $x_{i}$ is equal to some $y_{j}$. In other words, except for the re-ordering of elements in the decomposition, the above join is unique.

Title Kurosh-Ore theorem KuroshOreTheorem 2013-03-22 18:10:11 2013-03-22 18:10:11 CWoo (3771) CWoo (3771) 6 CWoo (3771) Theorem msc 06D05 msc 06C05 msc 06B05