Direct decompositions of orthomodular lattices
The authors characterize orthomodular lattices of the form \(D_ 1:\) \(B\times L_ 1\) (B Boolean, \(L_ 1\) without nonzero Boolean factor), \(D_ 2:\) \(B\times \prod_{j\in J}L_ j\) (B Boolean, \(L_ j\) simple, non-Boolean), \(FD_ 2\subseteq D_ 2:\) all the \(B\times L_ 1\times...\times L_ n\), n finite. Let L be an orthomodular lattice. Assume \(M\subseteq L\), \(C(M)=C(L)\), \(m=\bigvee \{c(F)|\) \(F\subseteq M\) finite\(\}\), exists, then L is of the form \(D_ 1\). Here, c(F) is Pulmannova's commutator. If L is complete, then L is of the form \(D_ 2\) iff every p-ideal of L which contains some c(F)\(\neq 0\) contains an atom of C(L). L is of the form \(FD_ 2\) iff every p-ideal of L generated by a set of commutators is principal. These results are used to show that the class of orthomodular lattices with non-zero modular homomorphic image is axiomatizable, closed under inverse homomorphic images, subalgebras and products. Some interesting examples and applications are given.
scientific article; zbMATH DE number 4177127
scientific article