Properties of theories whose algebras are free
A hard theorem of Givant classifies the finitary algebraic theories for which every algebra is free. From it, one easily deduces that all such theories are minimal (adding any further equations makes them trivial) and have a well-defined notion of dimension (free algebras on sets of different cardinality are never isomorphic). We prove both these properties directly, without the aid of Givant's theorem.
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