ISOLS AND GENERALIZED BOOLEAN-ALGEBRAS

Let r = (C, +, •) be a finite, hence atomic Boolean algebra. Then Γ is isomorphic to (Q, U, ∩), where Q is the family of all (finite) subsets of a (finite) set v, namely the set of all atoms of Γ. Moreover, if v has cardinality n, the Boolean algebra Γ is determined up to isomorphism by its order, i.e., 2n, or equivalently by the number n. We shall extend this theorem to atomic generalized Boolean algebras y = (C, +, •) in which the set C is isolated rather than finite. We have to impose some recursivity conditions on Γ which hold trivially, if Γ is finite. If these conditions are satisfied, Γ is effectively isomorphic to (Q, U, ∩), where Q is the family of all finite subsets of an isolated set v, namely the set of all atoms of Γ. Moreover, if v has RET (recursive equivalence type) N, the system Γ is determined up to effective isomorphism by its order, i.e., 2N, or equivalently by the RET N. This result is of some interest, since the role played in ordinary arithmetic by the family of all (finite) subsets of some finite set v is played in isolic arithmetic by the family of all finite subsets of some isolated set v. © 1990 Rocky Mountain Mathematics Consortium.