On Blocks in the Products and Ultraproducts of Orthomodular Lattices

Abstract

Let $\mathcal {OML}$ denote the class of orthomodular lattices (OMLs, quantum logics). Let L be an OML and let B be a maximal Boolean subalgebra of L. Then B is called a block of L. In the algebraic investigation of OMLs a natural question is whether the blocks of a product (resp. ultraproduct) of OMLs are products (resp. ultraproducts) of the blocks of the respective “coordinate” OMLs. We first add to the study of this question as regards the products and the centres of the products (a special mention deserves the result that the centre of the ultraproduct is the ultraproduct of the centres of the respective OMLs). Then we pass to the analogous questions for ultraproducts where we present main results of this note. Though this question on the “regular” behaviour of blocks in ultraproducts remains open in general, we provide a positive partial solution. This contributes to the understanding of varieties important to quantum theories – to the varieties that contain both set-representable OMLs and projection OMLs. We consider an axiomatizable class of the OMLs, $\mathcal {OML}_n$, whose blocks uniformly intersect in finite sets of the maximal cardinality of $2^n$. It is worth realizing within the connection to quantum logic theory that, for instance, the OMLs given by Greechie diagrams belong to $\mathcal {OML}_2$. The importance of the results is commented on in relation to the state space properties of OMLs.