Astroids and pastings (Q798342)

For a long time the loop lemma, due to Greechie, was the main tool for constructing finite orthomodular lattices. It was unsatisfactory that, already for height 4, the loop lemma did not cover all possible cases. This problem is solved by the theorem of the author on astroids and pastings in the present paper: For all chain-finite orthomodular lattices L the existence of a loop M of order 3 or 4 in L requires for M the existence of a central block or a central astroid, respectively. This condition is graphically easily recognizable. The technical condition of a pasted set of finite Boolean algebras \({\mathcal B}\) is introduced - where, for instance, for all B,\(C\in {\mathcal B}\) and \(m\in B\cap C\) there has to exist an astroid for m in \({\mathcal B}\). It is shown that the pasting L (set theoretic union of the elements) of \({\mathcal B}\) has naturally the structure of an orthomodular lattice and that every chain-finite orthomodular lattice L arises in this fashion. \({\mathcal B}\) is the set of blocks (maximal Boolean subalgebras) of L.