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- Local finiteness for Green's relations in semigroup varietiesPublication . Volkov, Mikhail; Silva, Pedro; Soares, FilipaA semigroup variety is said to be locally š¦-finite, where š¦ stands for any of Greenās relations ā, ā, ā, š, or š„, if every finitely generated semigroup in this variety has only finitely many š¦-classes. We characterize locally š¦-finite varieties of finite axiomatic rank in the language of āforbidden objectsā.