Some Finiteness Conditions for Orthomodular Lattices
Günter Bruns, Richard J. Greechie · Canadian Journal of Mathematics · 1982
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: An: L has at most n blocks, Bn: there exists a covering of L by at most n blocks, Cn: the set ﹛C(x)| x ∈ L﹜ has at most n elements, Dn: 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 ⇔ ∃ nAn, B ⇔ ∃ nBn, C ⇔ ∃ nCn and D ⇔ ∃ nDn.