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



Varieties with cofinal sets: examples and amalgamation

Authors: Peter Bruyns and Henry Rose
Journal: Proc. Amer. Math. Soc. 111 (1991), 833-840
MSC: Primary 08B10; Secondary 03C05, 03C20, 06B20, 08B25
MathSciNet review: 1039528
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Abstract: A variety $ \mathcal{V}$ has a cofinal set $ S \subset \mathcal{V}$ if any $ A \in \mathcal{V}$ is embeddable in a reduced product of members of $ S$. Amalgamation in and examples of such varieties are considered. Among other results, the following are proved: (i) every lattice is embeddable in an ultraproduct of finite partition lattices; (ii) if $ \mathcal{V}$ is a residually small, congruence distributive variety whose members all have one-element subalgebras, then the amalgamation class of $ \mathcal{V}$ is closed under finite products.

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

  • [1] J. L. Bell and A. B. Slomson, Models and ultraproducts, North-Holland, 1969. MR 0269486 (42:4381)
  • [2] C. Bergman, Amalgamation classes of some distributive varieties, Algebra Universalis 20 (1985), 143-166. MR 806610 (87g:08016)
  • [3] S. Burris and H. P. Sankapanavar, A course in universal algebra, Springer-Verlag, 1981. MR 648287 (83k:08001)
  • [4] C. Chang and H. J. Keisler, Model theory, North-Holland, 1973.
  • [5] G. Grätzer, General lattice theory, Birkhäuser Verlag, Basel and Stuttgart, 1978. MR 504338 (80c:06001a)
  • [6] G. Grätzer and H. Lakser, The structure of pseudocomplemented, distributive lattices II: Congruence extensions and amalgamations, Trans. Amer. Math. Soc. 156 (1971), 343-358. MR 0274359 (43:124)
  • [7] P. Jipsen and H. Rose, Absolute retracts and amalgamation in certain congruence-distributive varieties, Canad. Math. Bull. 32 (1989), 309-313. MR 1010069 (91b:08009)
  • [8] B. Jónsson, Amalgamation in small varieties of lattices, preprint, 1986.
  • [9] P. Pudlak and J. Tuma, Every finite lattice can be embedded in the lattice of all equivalences over a finite set, Algebra Universalis 10 (1980), 74-95. MR 552159 (81e:06013)
  • [10] W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33-53. MR 0314726 (47:3278)
  • [11] M. Yasuhara, The amalgamation property, the universal-homogeneous models and the generic models, Math. Scand. 34 (1974), 5-36. MR 0371642 (51:7860)

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Article copyright: © Copyright 1991 American Mathematical Society

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