   Then \(S\) is an isomorphism onto a BA of subsets of the set \(X\) of all ultrafilters on \(A\). This establishes the basic Stone representation theorem, and clarifies the origin of BAs as concrete algebras of sets. Moreover, the sets \(S(a)\) can be declared to be a base for a topology on \(X\), and this turns \(X\) into a totally disconnected compact Hausdorff space. This establishes a one-one correspondence between the class of BAs and the class of such spaces. As a consequence, used very much in the theory of BAs, many topological theorems and concepts have consequences for BAs. If \(x\) is an element of a BA, we let \(0x = -x\) and \(1x = x\). If \((x\)(0), \(\ldots x(m - 1))\) is a finite sequence of elements of a BA \(A\), then every element of the subalgebra of \(A\) generated by \(\{x(0), \ldots, x(m - 1)\}\) can be written as a sum of monomials \(e(0)x(0) \cdot \ldots \cdot e(m - 1)x(m - 1)\) for \(e\) in some set of functions mapping \(m = \{0,\ldots , m - 1\}\) into \(2 = \{0, 1\}\). This is an algebraic expression of the disjunctive normal form theorem of sentential logic. A function \(f\) from a set \(X\) of generators of a BA \(A\) into a BA \(B\) can be extended to a homomorphism if and only if