BOOLEAN SUBTRACTΓVΈ ALGEBRAS THOMAS M . HEARNE and

and prove that the study of such structures is equivalent to the study of generalized Boolean algebras. We call such structures Boolean subtractive algebras since they are subtractive algebras in the sense of Crapo and Rota ([1], 3.7). Alternatively, such structures might be called generalized colonies since, as we later prove, every colony is a Boolean subtractive algebra. As an example of a Boolean subtractive algebra which is not a colony, we mention the set of all finite subsets of an infinite set, with set difference as composition. This (infinite) model shows the consistency of our axioms. There are also many finite models of SI, S2, and S3, all of which are colonies. The independence of these axioms is also easily demonstrated. Let S be any two-element set and let x y = x for all x, ye S. This composition shows the independence of S3. If, on the other hand, one sets x y = y for