proceedings of the american mathematical society Volume 100, Number 4, August 1987 QUANTUM LOGICS WITH LATTICE STATE SPACES JIRÍ BINDER AND MIRKO NAVARA ABSTRACT. Let L be a quantum logic and let S(L) denote the set of all states on L. (By a state we mean a nonnegative bounded cr-additive mea- sure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10]. 1. Introduction and preliminaries. In axiomatic quantum mechanics (see [3, 8]), the "event structure" of a physical system is modelled by a cr-orthomodular partially ordered set (usually called a quantum logic). The states of the system then may be assumed to correspond to nonnegative bounded cr-additive measures on L, called "states" on L. Let us denote by S(L) the set of all states on L. If L is a Boolean cr-algebra, which is the case of an experiment subject to "classical mechanics," then S(L) is a lattice. In this paper we speculate whether this property of S(L) characterizes exactly "classical experiments." We find an axiomatic setup of the problem and obtain the results indicated in the abstract. Let us first review basic notions as we shall use them in the sequel. DEFINITION 1. A (quantum) logic is a set L equipped with a partial ordering < and a unary operation ', such that the following conditions are satisfied (the symbols A, V stand for the lattice operations induced by <): (1)0,16L, (2) o < b => tf < a' for any a, b G L, (3) (a1)' = a for any a G L, (4) if an (n G N) is a countable subset of L such that an < a'm whenever n^ m, then VngN an exists in L, and (5) if a, b G L and a < b then b = a V (b A a'). Obviously, a logic need not be distributive and need not be a lattice. Most frequently used examples of logics are Boolean er-algebras and lattices of projections in a von Neumann algebra. DEFINITION 2. A state on a logic L is a nonnegative bounded mapping s: L —> R such that (1) s(0) = 0, and (2) whenever an (n € A) is a sequence in L such that an < a'm for n/ m, then S(.Vn€Nan) = 2Zn€NS(an)- Received by the editors June 26, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 81B10, 06C15; Secondary 46E27, 81C20. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 688 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use