Stone's representation theorem for Boolean hyperalgebras. Topological version (Q1923496)

A Boolean hyperalgebra is, roughly speaking, obtained from a Boolean algebra \(A\) by replacing the disjunction by a hyperoperation \(\vee:A\times A\to{\mathcal P}(A)\) satisfying appropriate axioms; for example, absorption is replaced by \(a\in a\vee(a\wedge b)\). The standard example is the power set hyperalgebra \(({\mathcal P}(X),\vee, \cap,{}',\emptyset,X)\), where \(I\vee J=\{(I\setminus J)\cup(J\setminus I)\cup S: S\subseteq I\cap J\}\) for all \(I,J\in{\mathcal P}(X)\). Let \(\text{Ult}(A)\) denote the set of ultrafilters of \(A\), where the concept of ultrafilter is the conventional one. It turns out [the author and \textit{R. Rota}, Algebraic hyperstructures and applications, Proc. 4th Int. Congr., Xanthi 1990, 161-165 (1991; Zbl 0745.06004)] that the Stone map \(x\mapsto s(x)=\{p\in{\mathcal P}(\text{Ult} (A)):x\not\in p\}\) is a hyperalgebra embedding \(s:A\to{\mathcal P}(\text{Ult}(A))\), to the effect that \(s\) is injective and satisfies \(s(x\wedge y)=s(x)\wedge s(y)\) and \(s(x\vee y)\subseteq s(x)\vee s(y)\). Now the author shows that \(\text{Ult}(A)\) is a Boolean space having \(s(A)\) as clopen sets. He also proves that the identity \(s(x\vee y)=s(x)\vee s(y)\) holds if and only if \(A\) is finite.