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The variety (in the sense of universal algebra) of Boolean algebras, for example, has the property that finitely generated free algebra algebras have finite cardinality; in that case specifically $|F_n|=2^{2^n}$, in the obvious notation. Can one usefully characterize varieties whose finitely generated free algebras have finite cardinality? Can one characterize natural number sequences arising as $|F_n|$ in association with such varieties? |
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When finitely generated free algebras are free...finite |
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When finitely generated free algebras are free...The variety (in the sense of universal algebra) of Boolean algebras, for example, has the property that finitely generated free algebra have finite cardinality; in that case specifically $|F_n|=2^{2^n}$, in the obvious notation. Can one usefully characterize varieties whose finitely generated free algebras have finite cardinality? Can one characterize natural number sequences arising as $|F_n|$ in association with such varieties?
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