KuroshOre theorem
Theorem 1 (KuroshOre).
Let $L$ be a modular lattice^{} and suppose that $a\mathrm{\in}L$ has two irredundant decompositions of joins of joinirreducible elements:
$$a={x}_{1}\vee \mathrm{\cdots}\vee {x}_{m}={y}_{1}\vee \mathrm{\cdots}\vee {y}_{n}.$$ 
Then

1.
$m=n$, and

2.
every ${x}_{i}$ can be replaced by some ${y}_{j}$, so that
$$a={x}_{1}\vee \mathrm{\cdots}\vee {x}_{i1}\vee {y}_{j}\vee {x}_{i+1}\vee \mathrm{\cdots}\vee {x}_{m}.$$
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 reordering of elements in the decomposition, the above join is unique.
Title  KuroshOre theorem 

Canonical name  KuroshOreTheorem 
Date of creation  20130322 18:10:11 
Last modified on  20130322 18:10:11 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 06D05 
Classification  msc 06C05 
Classification  msc 06B05 