Kurosh-Ore theorem

Theorem 1 (Kurosh-Ore).

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

   $$a=x_1\vee \mathrm{\cdots }\vee x_{i-1}\vee y_j\vee x_{i+1}\vee \mathrm{\cdots }\vee x_m.$$

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 re-ordering of elements in the decomposition, the above join is unique.