Completions

One can sometimes establish a certain property for a class of relation algebras by applying the following strategy. First, prove that all complete and atomic relation algebras in the class possess the property, and then show that the property is inherited by subalgebras. If the class in question is closed under the formation of canonical extensions, that is to say, if the canonical extension of every algebra in the class also belongs to the class, then every algebra in the class will possess the desired property. Canonical extensions, however, do have a serious disadvantage from this perspective, because all properly infinite sums and products in the original algebra are changed in the passage to the canonical extension, and (in the infinite case) new atoms are introduced even when the original algebra is atomic. Consequently, if the property in question involves properly infinite sums or products, or atoms, in the original algebra, then the above strategy may fail. What one would like is an extension that is complete, with the same atoms as in the original algebra, and in which all existing infinite sums and products of the original algebra are preserved. Fortunately, such an extension exists, and in fact it satisfies a very nice minimality condition.