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

Samuel S. Holland
1970 Proceedings of the American Mathematical Society  
We call an orthomodular lattice £ m-orthocomplete for an infinite cardinal m if every orthogonal family of Sm elements from £ has a join in £, and we call £ m-complete if every family, orthogonal or not, of %m elements from £ has a join in £. We prove that an »z-orthocomplete orthomodular lattice is mcomplete. 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. © On-i -JA p + K 8)
more » ... ^ V (ya; a < 8). We need therefore prove only the statement P(B): V(ya; a<8) = © (yp+i-%; P + l<)3)-P(2) is the assertion yi^yi-yo which is true because yo = 0. We use transfinite induction. Assume thatPCy) Received by the editors June 30, 1969. A MS Subject Classifications. Primary 0635, 0660.
doi:10.1090/s0002-9939-1970-0256949-7 fatcat:gi5amomzjbcx3arapw6d6f2kfu