Boolean contact algebras

The origins of Boolean contact algebras go back to the works of Leśniewski [8] on mereology and Leonard and Goodman [7] on the calculus of individuals on the one hand, and, on the other hand, the efforts of e.g. de Laguna [2], Tarski [12] and Whitehead [13] to use regions instead of points as the basic entity of geometry. A central role played the notion of “connection” (or “contact”) of regions, which, in its simplest form, is a reflexive and symmetric relation C among non-empty regions, satisfying an additional extensionality axiom [3]. In order to formalize mereological structures (which were, basically, complete Boolean algebras B without a smallest element) together with Whitehead’s connection relation C, Clarke [1] proposed additional axioms, among them a compatibility axiom a ≤ b ⇐⇒ {u : aCu} ⊆ {u : bCu} and a summation axiom aC(u + v)⇐⇒ aCu or aCv. A subsequent development, the Region Connection Calculus (RCC) [10], additionally supposed that each proper non-zero region was connected to its complement. Boolean algebras of regular closed sets of regular T1 spaces, together with Whitehead’s connection aCb ⇐⇒ a∩ b 6= / 0 served as standard models for these "connection algebras". In a separate development, proximities on power set algebras were investigated which exhibited many similarities to the mereo-topological contact relations [6, 9, 11]. Three major questions arose: