## You are here

HomeKurosh-Ore theorem

## Primary tabs

# 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. $m=n$, and

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

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

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.

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

06D05*no label found*06C05

*no label found*06B05

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections