MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 15the projective algebras of Everett and Ulam [1946]; and the use of several S5operators to provide a Boolean model of features of first-order logic.3.3 BAO’s: The Theory of Jónsson and TarskiThe notion of a Boolean algebra with operators (BAO) was introduced by Jónssonand Tarski in their abstract [1948], with the details of their announced results beingpresented in [1951]. That work contains representations of algebras that couldimmediately have been applied to give new characterisations of modal systems.But the paper was overlooked by modal logicians, who were still publishing rediscoveriesof some of its results fifteen years later.A unary function f on a Boolean algebra is an operator if it is additive, i.e.f(x + y) = f(x) + f(y). f is completely additive if f( ∑ X) = ∑ ∑f(X) wheneverX exists, and is normal if f(0) = 0. A function of more than one argumentis an operator/is completely additive/is normal when it is has the correspondingproperty separately in each argument. A BAO is an algebra A = (B, f i : i ∈ I),where the f i ’s are all operators on the Boolean algebra B.The Extension Theorem of Jónsson and Tarski showed that any BAO A can beembedded isomorphically into a complete and atomic BAO A σ which they called aperfect extension of A. The construction built on Stone’s embedding of a Booleanalgebra B into a complete and atomic one B σ , with each operator f i of A beingextended to an operator fiσ on B σ that is completely additive, and is normal if f iis normal. The notion of perfect extension was defined by three properties thatdetermine A σ uniquely up to a unique isomorphism over A and give an algebraiccharacterisation of the structures that arise from Stone’s topological representationtheory. These properties can be stated as follows.(i)(ii)For any distinct atoms x, y of A σ there exists an element a of A with x ≤ aand y ≤ −a.If a subset X of A has ∑ X = 1 in A σ , then some finite subset X 0 of X has∑X0 = 1.(iii)f σ i (x) = ∏ {f i (y) : x ≤ y ∈ A n } when f i is n-ary and the terms of then-tuple x are atoms or 0.Property (i) corresponds to the Hausdorff separation property of the Stone spaceof B, while (ii) is an algebraic formulation of the compactness of that space. Themeaning of (iii) will be explained below.Jónsson and Tarski showed that any equation satisfied by A will also be satisfiedby A σ if it does not involve Boolean complementation (i.e. refers only to +, ·, 0, 1and the operators f i ). More generally, perfect extensions were shown to preserveany implication of the form (t = 0 → u = v) whose terms t, u, v do not involvecomplementation. They then established a fundamental representation of normal