A semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ℋ, ℛ, ℒ, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.
