An Algebraic Characterization of Semirings for Which the Support of Every Recognizable Series Is Recognizable

We show that for a semiring ${\mathbb K}$, the support of all recognizable series over ${\mathbb K}$ are recognizable if and only if in every finitely generated subsemiring of ${\mathbb K}$ there exists a finite congruence such that $0_{\mathbb K}$ is a singleton congruence class.

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