Projective orthomodular lattices. II (Q1272097)

The authors continue the study of projectivity in orthomodular lattices started in Part I [Can. Math. Bull. 37, No. 2, 145-153 (1994; Zbl 0819.06007)]. The main results: Theorem 1.1. No uncountable Boolean algebra is projective in the variety of all orthomodular lattices. Corollary 1.3. Every Boolean subalgebra of a free orthomodular lattice is at most countable. Theorem 1.4. Let \(K\) be a class of algebras which have a semillatice reduct. If \(F\) is free in \(K\) and if \(F\) has no countably generated uncountable subalgebra then every chain in \(F\) is at most countable. Corollary 1.5. Every chain in a free orthomodular lattice is at most countable.