An abelian Mal'cev variety is finitely based exactly when it has finite type and its ring of idempotent binary terms and module of unary terms are finitely presented.
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Characterizing Finitely Based Abelian Mal'cev Algebras
An abelian Mal'cev variety is finitely based exactly when it has finite type and its ring of idempotent binary terms and module of unary terms are finitely presented.