Equational theories with a minority polynomial

Padmanabhan, R.; Wolk, B.

doi:10.1090/S0002-9939-1981-0624905-X

Equational theories with a minority polynomial
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by R. Padmanabhan and B. Wolk PDF

Proc. Amer. Math. Soc. 83 (1981), 238-242 Request permission

Abstract:

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.

References

Kirby A. Baker, Finite equational bases for finite algebras in a congruence-distributive equational class, Advances in Math. 24 (1977), no. 3, 207–243. MR 447074, DOI 10.1016/0001-8708(77)90056-1
Bjarni Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121 (1968). MR 237402, DOI 10.7146/math.scand.a-10850
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Ralph McKenzie, On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model, J. Symbolic Logic 40 (1975), 186–196. MR 376323, DOI 10.2307/2271899
R. Padmanabhan and R. W. Quackenbush, Equational theories of algebras with distributive congruences, Proc. Amer. Math. Soc. 41 (1973), 373–377. MR 325498, DOI 10.1090/S0002-9939-1973-0325498-2
R. Padmanabhan, Equational theory of algebras with a majority polynomial, Algebra Universalis 7 (1977), no. 2, 273–275. MR 434924, DOI 10.1007/BF02485437
A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963), 105–109. MR 146104, DOI 10.1090/S0002-9939-1963-0146104-X
Walter Taylor, Equational logic, Houston J. Math. Survey (1979), iii+83. MR 546853