Isols and generalized Boolean algebras

Abstract

Let Γ = C, +, • be a finite, hence atomic Boolean algebra.Then Γ is isomorphic to Q, ∪, ∩ , where Q is the family of all (finite) subsets of a (finite) set ν, namely the set of all atoms of Γ.Moreover, if ν has cardinality n, the Boolean algebra Γ is determined up to isomorphism by its order, i.e., 2 n , or equivalently by the number n.We shall extend this theorem to atomic generalized Boolean algebras Γ = 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, ∪, ∩ , where Q is the family of all finite subsets of an isolated set ν, namely the set of all atoms of Γ.Moreover, if ν has RET (recursive equivalence type) N , the system Γ is determined up to effective isomorphism by its order, i.e., 2 N , 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 ν is played in isolic arithmetic by the family of all finite subsets of some isolated set ν.1. Algebraic preliminaries.Let Δ = D, +, • be a distributive lattice.For u, v ∈ D we often abbreviate "u • v" to "uv."The distributive lattice Δ has a zero-element if there is an element 0 ∈ D such that x + 0 = x for all x ∈ D. Similarly, Δ has a one-element if there is an element 1 ∈ D such that x • 1 = x for all x ∈ D. If p, q ∈ D we define p ≤ q as pq = p or equivalently as p + q = q.For a, b ∈ D we write [a, b]there are elements a, b ∈ D such that a ≤ b and S = [a, b].Note that, for a, b, p, q ∈ D, (a ≤ p ≤ b and a ≤ q ≤ b) ⇒ (a ≤ p + q ≤ b and a ≤ pq ≤ b), i.e., that [a, b] is closed under + and • .Thus, by restricting the operations + and • of Δ to the interval [a, b] we obtain a distributive lattice with a as zero-element and b as one-element.This is called the lattice induced by Δ in [a, b].