An Extension of Greechie's Atomistic Loop Lemma

Abstract

<p>One of the major deficiencies of the theory of orthomodular lattices is the lack of accessible examples with which to test and develop conjectures. R. J. Greechie has developed two methods ([1], [2]) of obtaining new orthmodular lattices by "pasting" old ones together. This thesis gives an extension of one of these, the atomistic loop lemma. Briefly, a non-empty set of Boolean lattices with common bounds which either intersect trivially or in disjoint principal sections form an orthocomplemented poset under set-theoretic union. Necessary and sufficient conditions are given for this poset to be orthomodular.</p>