The Structure of Endomorphism Monoids in Conjugate Categories

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International Journal of Algebra

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We invoke structure theorems from the theory of semigroups to analyze the anatomy of categories arising from a type of conjugation by a subcategory. The equivalence classes of Green’s H-relation admit group structures when they contain idempotents, and we use this to study the endomorphism monoids of the objects in a conjugate category. We prove that in any such category, the H-group associated to an idempotent endomorphism is always realizable as the automorphism group of its “rank object,” which serves as a generalized image. As all H-groups are automorphism groups of lower rank, we establish a resulting homogeneity of these H-groups, giving stringent necessary conditions for the existence of conjugate categories.


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