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)



Two observations on the congruence extension property

Authors: G. Grätzer and H. Lakser
Journal: Proc. Amer. Math. Soc. 35 (1972), 63-64
MSC: Primary 08A25
MathSciNet review: 0297677
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A pair of algebras $ \mathfrak{U},\mathfrak{B}$ with $ \mathfrak{B}$ a subalgebra of $ \mathfrak{U}$ is said to have the (Principal) Congruence Extension Property (abbreviated as PCEP and CEP, respectively) if every (principal) congruence relation of $ \mathfrak{B}$ can be extended to $ \mathfrak{U}$. A pair of algebras $ \mathfrak{U}$, $ \mathfrak{B}$ is constructed having PCEP but not CEP, solving a problem of A. Day. A result of A. Day states that if $ \mathfrak{B}$ is a subalgebra of $ \mathfrak{U}$ and if for any subalgebra $ \mathfrak{C}$ of $ \mathfrak{U}$ containing $ \mathfrak{B}$, the pair $ \mathfrak{U},\mathfrak{C}$ has PCEP, then $ \mathfrak{U},\mathfrak{B}$ has CEP. A new proof of this theorem that avoids the use of the Axiom of Choice is also given.

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

  • [1] A. Day, A note on the Congruence Extension Property, Algebra Universalis 1 (1971), 234-235. MR 0294215 (45:3288)
  • [2] E. Fried and G. Grätzer, A nonassociative extension of the class of distributive lattices. I, II, Notices Amer. Math. Soc. 18 (1971), 402, 548. Abstract #71T-A47; #71T-A62.
  • [3] A. W. Goldie, The Jordan-Hölder theorem for general abstract algebras, Proc. London Math. Soc. (2) 52 (1950), 107-131. MR 12, 238. MR 0037289 (12:238e)
  • [4] G. Grätzer, On the Jordan-Hölder theorem, for universal algebras, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 397-406. MR 29 #4717. MR 0167444 (29:4717)
  • [5] -, Universal algebra, Van Nostrand, Princeton, N.J., 1968. MR 40 #1320. MR 0248066 (40:1320)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 08A25

Retrieve articles in all journals with MSC: 08A25

Additional Information

Keywords: Universal algebra, congruence relation, Congruence Extension Property
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society