Connection Structures

B. L. Clarke, following a proposal of A. N. Whitehead, presents an axiomatized calculus of individuals based on a primitive predicate "x is connected with y". In this article we show that a proper subset of Clarke's system of axioms characterizes the complete orthocomplemented lattices, while the whole of Clarke's system characterizes the complete atomless Boolean algebras. / Introduction In [2] and [3] Clarke presents an axiomatized calculus of individuals based on a primitive predicate "x is connected with y". Such a calculus represents a revised version of the proposal made by Whitehead in Process and Reality and is similar to the calculus proposed by Leonard and Goodman in [5]. In this article we show that a proper subset of Clarke's system of axioms characterizes the complete orthocomplemented lattices, while the whole of Clarke's system characterizes the complete atomless Boolean algebras. 2 Connection structures Let R be a nonempty set and C a binary relation on R, set C(x) = [y G R/xCy] and suppose the following axioms are true of every x9y E R: Al xCx; A2 xCy => yCx; A3 C(x) = C(y)=>x = y. We call regions the elements of R and, if x,y G R and xCy, we say that x is connected with y. If X is a nonempty subset of R, we say that x is the fusion of X just in case for every y ELR, xCy iff for some zGX, zCy; in other words, x is the fusion of x provided that (1) C{x)=U[C{z)/zeX). The fusion of the nonempty subsets of R is assured by the following axiom. Received August 5, 1988; revised December 15, 1988 CONNECTION STRUCTURES 243 A4 X g R and X Φ 0 =» there exists x E R such that x is the fusion of X. If A1-A4 are satisfied, we say that (R = (R,C) is a connection structure. By A3 and A4 there is a unique fusion of a nonempty class X of regions; we denote it by f(X). A3 implies that the relation < defined in R by