An $m$-orthocomplete orthomodular lattice is $m$-complete

Abstract:

We call an orthomodular lattice $\mathcal {L}\;m$-orthocomplete for an infinite cardinal $m$ if every orthogonal family of $\leqq m$ elements from $\mathcal {L}$ has a join in $\mathcal {L}$, and we call $\mathcal {L}\;m$-complete if every family, orthogonal or not, of $\leqq m$ elements from $\mathcal {L}$ has a join in $\mathcal {L}$. We prove that an $m$-orthocomplete orthomodular lattice is $m$-complete. Since a Boolean algebra is a distributive orthomodular lattice, we obtain as a special case the Smith-Tarski theorem: An $m$-orthocomplete Boolean algebra is $m$-complete.