Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Equational theories with a minority polynomial

Authors: R. Padmanabhan and B. Wolk
Journal: Proc. Amer. Math. Soc. 83 (1981), 238-242
MSC: Primary 08B05; Secondary 20M05
MathSciNet review: 624905
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] K. A. Baker, Finite equational bases for finite algebras in a congruence-distributive equational class, Adv. in Math. 24 (1977), 207-243. MR 0447074 (56:5389)
  • [2] B. Jonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121. MR 0237402 (38:5689)
  • [3] R. C. Lyndon and P. E. Schupp, Combinational group theory, Springer-Verlag, New York, 1977. MR 0577064 (58:28182)
  • [4] R. N. 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 0376323 (51:12499)
  • [5] R. Padmanabhan and R. Quackenbush, Equational theories of algebras with distributive congruences, Proc. Amer. Math. Soc. 41 (1973), 373-377. MR 0325498 (48:3845)
  • [6] R. Padmanabhan, Equational theory of algebras with a majority polynomial, Algebra Universalis 7 (1977), 273-275. MR 0434924 (55:7887)
  • [7] A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963), 105-109. MR 0146104 (26:3630)
  • [8] W. Taylor, Equational logic, Appendix 4 in G. Grätzer, Universal algebra, 2nd ed., Springer-Verlag, New York, 1979. MR 546853 (80j:03042)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 08B05, 20M05

Retrieve articles in all journals with MSC: 08B05, 20M05

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society