Ever since Stone represented Boolean algebras topologically [Stone36], many have extended his theorem to diverse algebraic systems. Structurally, it suffices to discover a fragment of the congruence lattice that is Boolean. The value of such a fragment, to be called a Boolean subsemilattice in the first section, is flexibility. By varying the Boolean subsemilattice to suit the context, different representation theorems follow automatically. In a later chapter, for example, by looking at all the factor ideals of a unital ring in which the annihilator of any element is a principal ideal generated by an idempotent, we obtain stalks with no zero divisors. With sheaves this theorem can be extended well beyond ring theory.