REGULARITY IN QUANTUM LOGIC

In 1996, Harding showed that the binary decompositions of any algebrai c, relational, or topological structure X form an orthomodular poset F act X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompositions in Fact X is compatible if and only if all the binary decompositions in S can be built from a common n-ary decom position of X. This characterization of compatibility is used to show that for any algebraic, relational, or topological structure X, the or thomodular poset Fact X is regular. Special cases of this result inclu de the known facts that the orthomodular posets of splitting subspaces of an inner product space are regular, and that the orthomodular pose ts constructed from the idempotents of a ring are regular. This result also establishes the regularity of the orthomodular posets that Musht ari constructs from bounded modular lattices, the orthomodular posets one constructs from the subgroups of a group, and the orthomodular pos ets one constructs from a normed group with operators. Moreover, all t hese orthomodular posers are regular for the same reason. The characte rization of compatibility is also used to show that for any structure X, the finite Boolean subalgebras of Fact X correspond to finitary dir ect product decompositions of the structure X. For algebraic and relat ional structures X, this result is extended to show that the Boolean s ubalgebras of Fact X correspond to representations of the structure X as the global sections of a sheaf of structures over a Boolean space. The above results can be given a physical interpretation as well. Assu me that the true or false questions Q of a quantum mechanical system c orrespond to binary direct product decompositions of the state space o f the system, as is the case with the usual von Neumann interpretation of quantum mechanics. Suppose S is a subset of Q. Then a necessary an d sufficient condition that all questions in S can bd answered simulta neously is that any two questions in S can be answered simultaneously. Thus, regularity in quantum mechanics follows from the assumption tha t questions correspond to decompositions.