Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:
A
n
: L has at most n blocks,
B
n
: there exists a covering of L by at most n blocks,
C
n: the set ﹛C(x)| x ∈ L﹜ has at most n elements,
D
n
: out of any n + 1 elements of L at least two commute.
We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ n
An
, B ⇔ ∃ n
Bn
, C ⇔ ∃ n
Cn
and D ⇔ ∃ n
Dn
.