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)



Finite simple abelian algebras are strictly simple

Author: Matthew A. Valeriote
Journal: Proc. Amer. Math. Soc. 108 (1990), 49-57
MSC: Primary 08A30; Secondary 03C05
MathSciNet review: 990434
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term $ t(x,\bar y)$ and for all elements $ a,b,\bar c,\bar d$, we have the following implication: $ t(a,\bar c) = t(a,\bar d) \to t(b,\bar c) = t(b,\bar d)$. It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a well-known fact about Abelian groups and modules.

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

  • [1] Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Springer-Verlag, New York, 1981. MR 648287 (83k:08001)
  • [2] B. Csákány, Abelian properties of primitive classes of universal algebras, Acta. Sci. Math. (Szeged) 25 1964, 202-208. MR 0169805 (30:48)
  • [3] Ralph Freese and Ralph McKenzie, Commutator theory for congruence modular varieties, Vol. 125, London Math. Soc. Lec. Note Series, Cambridge University Press, 1987. MR 909290 (89c:08006)
  • [4] David Hobby and Ralph McKenzie, The structure of finite algebras, Vol. 76, Contemp. Math., American Mathematical Society, Providence, Rhode Island, 1988. MR 958685 (89m:08001)
  • [5] Emil Kiss and Matthew Valeriote, Strongly Abelian varieties and the Hamiltonian property (preprint), 1989. MR 1113758 (92g:08004)
  • [6] Ralph McKenzie, Finite forbidden lattices, in Universal Algebra and Lattice Theory, Vol. 1004, Springer Lecture Notes, Springer-Verlag, New York, 1983. MR 716183 (85b:06006)
  • [7] -, Congruence extension, Hamiltonian and Abelian properties in locally finite varieties (preprint), 1989.
  • [8] Ralph McKenzie and Matthew Valeriote, The structure of locally finite decidable varieties, Birkhäuser, Boston, 1989. MR 1033992 (92j:08001)
  • [9] P. P. Pálfy, Unary polynomials in algebras, Algebra Universalis, 18 (1984) 262-273. MR 745492 (86h:08001a)
  • [10] P. P. Pálfy and P. Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), 22-27. MR 593011 (82g:08003)
  • [11] K. Shoda, Zur theorie der algebraischen erweiterungen, Osaka J. Math. 4 (1952), 133-143. MR 0052403 (14:614b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 08A30, 03C05

Retrieve articles in all journals with MSC: 08A30, 03C05

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society