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Weak additive category. (Chinese. English summary) Zbl 0987.18010
Summary: We introduce the concept of a weak additive category based on the category $${\mathcal A}bs$$ of commutative monoids and the one $${\mathcal A}bs^M$$ of semimodules and discuss some properties of the coproducts of finite objects in them. Two necessary and sufficient conditions of that a functor of the category to itself reserves finite coproducts are given. Further, we introduce the concepts of ideals, congruences of a weak additive category and, based on them, prove the fundamental theorem of homomorphisms of a weak additive category and the first and second isomorphism theorem. Finally, we also discuss the lattice of congruences and subdirect products of weak additive categories. These concepts and results generalize that of additive categories and make a base of researches on the construction of weak additive categories.
##### MSC:
 18E05 Preadditive, additive categories 16Y99 Generalizations 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)