Journal of Pure and Applied Algebra (2025) 229(1)

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Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon “filtered colimit elimination,” and study a sufficient condition for it. We show that if a locally finitely presentable category A satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to A.

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APA

Kawase, Y. (2025). Filtered colimit elimination from Birkhoff’s variety theorem. *Journal of Pure and Applied Algebra*, *229*(1). https://doi.org/10.1016/j.jpaa.2024.107794

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