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Closedness properties of internal relations. IV: Expressing additivity of a category via subtractivity. (English) Zbl 1124.18003

[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.

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)

Citations:

Zbl 0799.08010
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