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Proceedings of the American Mathematical Society

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

 
 

 

Extending congruence relations


Author: Peter Krauss
Journal: Proc. Amer. Math. Soc. 31 (1972), 517-520
DOI: https://doi.org/10.1090/S0002-9939-1972-0285470-7
MathSciNet review: 0285470
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Abstract | References | Additional Information

Abstract: If $ \mathfrak{A}$ and $ \mathfrak{B}$ are algebras, where $ \mathfrak{A} \subseteq \mathfrak{B}$, and $ \theta $ is a congruence relation on $ \mathfrak{A}$, let $ {\theta ^\mathfrak{B}}$ be the smallest congruence relation on $ \mathfrak{B}$ containing $ \theta $. $ \mathfrak{A}$ is called a congruence subalgebra of $ \mathfrak{B}$ if $ \mathfrak{A} \subseteq \mathfrak{B}$ and, for every congruence relation $ \theta $ on $ \mathfrak{A},{\theta ^\mathfrak{B}} \cap \vert\mathfrak{A}{\vert^2} = \theta $. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of ``subalgebra'' is everywhere replaced by ``congruence subalgebra".


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  • [1] T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. Amer. Math. Soc. 3 (1952), 614-620. MR 14, 347. MR 0050566 (14:347h)
  • [2] -, On multiplicative systems defined by generators and relations. II, Proc. Cambridge Philos. Soc. 49 (1953), 579-589. MR 15, 283. MR 0057865 (15:283h)
  • [3] I. Fleischer, A note on universal homogeneous models, Math. Scand. 19 (1966), 183-184. MR 35 #4095. MR 0213231 (35:4095)
  • [4] G. Grätzer, Universal algebra, Van Nostrand, Princeton, N.J., 1968. MR 40 #1320. MR 0248066 (40:1320)
  • [5] G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247-254. MR 11, 322. MR 0032641 (11:322d)
  • [6] B. Jónsson, Universal relational systems, Math. Scand. 4 (1956), 193-208. MR 20 #3091. MR 0096608 (20:3091)
  • [7] -, Homogeneous universal relational systems, Math. Scand. 8 (1960), 137-142. MR 23 #A2328. MR 0125021 (23:A2328)
  • [8] A. I. Mal'cev, On a representation of nonassociative rings, Uspehi Mat. Nauk 7 (1952), no. 1 (47), 181-185. MR 13, 816. MR 0047028 (13:816e)
  • [9] M. Morley and R. Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37-57. MR 27 #37. MR 0150032 (27:37)
  • [10] A. Robinson, A result on consistency and its application to the theory of definition, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 47-58. MR 17, 1172. MR 0078307 (17:1172d)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0285470-7
Keywords: Congruence relation, elementary subalgebra, Directed Union Theorem, Loewenheim-Skolem Theorem, embedding property, amalgamation property, homogeneous-universal algebra
Article copyright: © Copyright 1972 American Mathematical Society

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