Quantum Logics Defined by Divisibility Conditions

Abstract

Let p be a prime number and let S be a countable set. Let us consider the collection DivpS of all subsets of S whose cardinalities are multiples of p and the complements of such sets. Then the collection DivpS constitutes a (set-representable) quantum logic (i.e., DivpS is an orthomodular poset). We show in this note that each state on DivpS can be extended over the Boolean algebra expS of all subsets of S. We also prove that all pure states on DivpS are two-valued. (If we lend to a main result a possible interpretation in terms of quantum entities, the logics DivpS have higher degree of noncompatibility but somewhat classical states.)