In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze’s notion of semidirect product in ideally exact categories is analysed.
Keywords: Ideally exact category, semi-abelian category, internal action, non-associative algebra, MV-algebra, product algebra
2020 MSC: 03B52; 06D35; 08C05; 16B50; 18C05; 18E13
Theory and Applications of Categories, Vol. 45, 2026, No. 31, pp 1280-1320.
Published 2026-05-22.
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