Janelidze, Zurab Closedness properties of internal relations. IV: Expressing additivity of a category via subtractivity. (English) Zbl 1124.18003 J. Homotopy Relat. Struct. 1, No. 1, 219-227 (2006). [For part I–III, see the preceding review.]A variety \(V\) of universal algebras is subtractive in the sense of A. Ursini [Algebra Univers. 31, No. 2, 204–222 (1994; Zbl 0799.08010)], 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 \(s(x,x)=0.\) 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. In this paper the author shows that a subtractive variety \({\mathcal C}\), whose theory contains a unique constant, is abelian (i.e., \({\mathcal C}\) is the variety of modules over a fixed ring), if and only if the dual category \({\mathcal C}^{\text{op}}\) of \({\mathcal C}\) is subtractive. More generally, he shows that \({\mathcal C}\) is additive if and only if both \({\mathcal C}\) and \({\mathcal C}^{\text{op}}\) are subtractive, where \({\mathcal C}\) is an arbitrary finitely complete pointed category, with binary sums, and such that each morphism \(f\) in \({\mathcal C}\) can be presented as a composite \(f=me,\) where \(m\) is a monomorphism and \(e\) is an epimorphism. Reviewer: Dana Piciu (Craiova) Cited in 2 ReviewsCited in 5 Documents MSC: 18C99 Categories and theories 18E05 Preadditive, additive categories 18E10 Abelian categories, Grothendieck categories 08B05 Equational logic, Mal’tsev conditions 08C05 Categories of algebras 18D35 Structured objects in a category (MSC2010) Keywords:subtractive category; subtraction algebra; closedness property Citations:Zbl 0799.08010 PDFBibTeX XMLCite \textit{Z. Janelidze}, J. Homotopy Relat. Struct. 1, No. 1, 219--227 (2006; Zbl 1124.18003) Full Text: arXiv EuDML EMIS