An equational theory is a theory axiomatized by a set of equations. Does every such theory have an independent equational axiomatization? Independent means no axiom in the set can be deleted without loss of theorems.
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Two quick comments. First, note that the "obvious" argument of enumerating the original theory and recursively selecting "not-yet-provable" axioms does not work since later axioms may prove earlier axioms, and throwing out old axioms may result in the "limit axiomatization" being too weak. Second, note that this old answer of mine is a red herring: even if the original axiomatization is equational, the new axioms will not be equations but rather implications whose conclusion is an equation and whose hypothesis is a conjunction of equations. – Noah Schweber Jul 25 '20 at 16:53
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No.
Finite algebras with no independent basis of identities,
I. M. ISAEV,
Algebra univers. 37 (1997) 440-444
describes a finite algebra whose equational theory has no independent equational axiomatization. The algebra is a finite dimensional vector space over a finite field equipped with a certain nonassociative bilinear multiplication.
Keith Kearnes
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