Kurosh-Ore theorem

Theorem 1 (Kurosh-Ore).

Let L be a modular latticeMathworldPlanetmath and suppose that aL has two irredundant decompositions of joins of join-irreducible elements:



  1. 1.

    m=n, and

  2. 2.

    every xi can be replaced by some yj, so that


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

Remark. Additionally, if L is a distributive latticeMathworldPlanetmath, then the second property above (known the replacement property) can be strengthened: each xi is equal to some yj. In other words, except for the re-ordering of elements in the decomposition, the above join is unique.

Title Kurosh-Ore theorem
Canonical name KuroshOreTheorem
Date of creation 2013-03-22 18:10:11
Last modified on 2013-03-22 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