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 has a cofinal set if any is embeddable in a reduced product of members of . 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 is a residually small, congruence distributive variety whose members all have one-element subalgebras, then the amalgamation class of is closed under finite products.

**[1]**J. L. Bell and A. B. Slomson,*Models and ultraproducts: An introduction*, North-Holland Publishing Co., Amsterdam-London, 1969. MR**0269486****[2]**Clifford Bergman,*Amalgamation classes of some distributive varieties*, Algebra Universalis**20**(1985), no. 2, 143–166. MR**806610**, 10.1007/BF01278593**[3]**Stanley Burris and H. P. Sankappanavar,*A course in universal algebra*, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR**648287****[4]**C. Chang and H. J. Keisler,*Model theory*, North-Holland, 1973.**[5]**George Grätzer,*General lattice theory*, Birkhäuser Verlag, Basel-Stuttgart, 1978. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. MR**504338****[6]**G. Grätzer and H. Lakser,*The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation*, Trans. Amer. Math. Soc.**156**(1971), 343–358. MR**0274359**, 10.1090/S0002-9947-1971-0274359-9**[7]**Peter Jipsen and Henry Rose,*Absolute retracts and amalgamation in certain congruence distributive varieties*, Canad. Math. Bull.**32**(1989), no. 3, 309–313. MR**1010069**, 10.4153/CMB-1989-044-x**[8]**B. Jónsson,*Amalgamation in small varieties of lattices*, preprint, 1986.**[9]**Pavel Pudlák and Jiří Tuma,*Every finite lattice can be embedded in a finite partition lattice*, Algebra Universalis**10**(1980), no. 1, 74–95. MR**552159**, 10.1007/BF02482893**[10]**Walter Taylor,*Residually small varieties*, Algebra Universalis**2**(1972), 33–53. MR**0314726****[11]**Mitsuru Yasuhara,*The amalgamation property, the universal-homogeneous models, and the generic models*, Math. Scand.**34**(1974), 5–36. MR**0371642**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1039528-7

Article copyright:
© Copyright 1991
American Mathematical Society