Closedness Properties of Internal Relations Iv: Expressing Additivity of a Category via Subtractivity

The notion of a subtractive category, recently introduced by the author, is a “categorical version” of the notion of a (pointed) subtractive variety of universal algebras, due to A.Ursini. We show that a subtractive variety C, whose theory contains a unique constant, is abelian (i.e. C is the variety of modules over a fixed ring), if and only if the dual category Cop of C, is subtractive. More generally, we show that C is additive if and only if both C and Cop are subtractive, where C is an arbitrary finitely complete pointed category, with binary sums, and such that each morphism f in C can be presented as a composite f = me, where m is a monomorphism and e is an epimorphism. Introduction A variety V of universal algebras is subtractive in the sense of A. Ursini [14], if the theory of V contains a binary term s (called a subtraction term) and a nullary term 0, satisfying the identities s(x, 0) = x and s(x, x) = 0. An example of a subtractive variety is the variety of (additive) groups, for which s(x, y) = x−y. More generally, any Mal’tsev variety [12], whose theory contains a nullary term, is subtractive. An example of a subtractive variety, which is not a Mal’tsev variety, is the variety of implication algebras [1]. A subtraction algebra is a triple A = (A,−, 0), where A is a set, “−” is a binary operation on A, and 0 is a nullary operation on A, satisfying the axioms x− 0 = x and x− x = 0. A group can be defined as a subtraction algebra with the additional axiom (x−y)−(z−y) = x−z, and an abelian group can be defined as a subtraction algebra with the stronger axiom (x− y)− (z − t) = (x− z)− (y − t); (1) Partially supported by South African National Research Foundation. Received July 19, 2006, revised September 21, 2006; published on October 12, 2006. 2000 Mathematics Subject Classification: 18E05, 18E10, 18C99, 08B05, 08C05, 18D35.