Chu spaces: Complementarity and Uncertainty in Rational Mechanics

A Boolean Chu space A = (X, |=, A) consists of two sets X and A and a binary relation |= ⊆ X × A from X to A. We call the elements x, y, . . . of X states or opens, and the elements a, b, . . . of A points, propositions, or events. We read x |= a as the Boolean-valued assertion “state x satisfies point (proposition, event) a.” Viewed as an event, a is understood as the proposition “event a has happened.” A Chu space can be depicted naturally as a matrix. Figure 1 gives some illustrative examples that we shall refer to in the sequel. We define rowA(x) = {a | x |= a}, the set of those column indices a containing a 1 at row x, and dually colA(a) = {x | x |= a}. When rowA is an injective function (no repeated rows) we call A extensional, and when colA is injective (no repeated columns) we call A T0 by analogy with topological spaces. When rowA is the identity function on X, X must be a subset of 2; we call such a Chu space normal, and write it as simply (X,A), |= then being inferrable as the converse ∈ ̆ of set membership. A normal space is automatically extensional but need not be T0. The Chu spaces of Figure 1 are all extensional and T0 but ∗This work was supported by ONR under grant number N00014-92-J-1974.