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.
2026-03-29 02:17:29.1774750649
Does every equational theory have an independent equational axiomatization?
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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.