Equational theories with a minority polynomial
Ranganathan Padmanabhan, Barry Wolk · Proceedings of the American Mathematical Society · 1981
It is known that every finitely based variety of algebras with distributive and permutable congruences is one-based and those admitting a majority polynomial are two-based. In this note we prove two results, one similar to the above and the other in a completely opposite direction: (i) every finitely based variety of algebras of type $\left \langle 3 \right \rangle$ satisfying the two-thirds minority condition is one-based and (ii) for every natural number $n$, there exists a variety of algebras admitting even a full minority polynomial which is $(n + 1)$-based but not $n$-based. An application to the strict consistency of defining relations for semigroups is given.