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Maximal $n$ -generated subdirect products

Joel Berman

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL, USA

E-mail Address: jberman@uic.edu

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https://doi.org/10.1142/S0218196716500065Cited by:0 (Source: Crossref)

Abstract

For $n$ a positive integer and $K$ a finite set of finite algebras, let $L (n, K)$ denote the largest $n$ -generated subdirect product whose subdirect factors are algebras in $K$ . When $K$ is the set of all $n$ -generated subdirectly irreducible algebras in a locally finite variety $𝒱$ , then $L (n, K)$ is the free algebra $F_{𝒱} (n)$ on $n$ free generators for $𝒱$ . For a finite algebra $A$ the algebra $L (n, {A})$ is the largest $n$ -generated subdirect power of $A$ . For every $n$ and finite $A$ we provide an upper bound on the cardinality of $L (n, {A})$ . This upper bound depends only on $n$ and these basic parameters: the cardinality of the automorphism group of $A$ , the cardinalities of the subalgebras of $A$ , and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of $A$ . Using this upper bound on $n$ -generated subdirect powers of $A$ , as $A$ ranges over the $n$ -generated subdirectly irreducible algebras in $𝒱$ , we obtain an upper bound on $| F_{𝒱} (n) |$ . And if all the $n$ -generated subdirectly irreducible algebras in $𝒱$ have congruence lattices that are chains, then we characterize in several ways those $𝒱$ for which this upper bound is obtained.

Communicated by K. Kearnes

Keywords:

AMSC: 08A40, 08B20, 08B26, 03C05, 05A15, 68Q25

References

1. H. Andréka, B. Jónsson and I. Németi, Free algebras in discriminator varieties, Algebra Universalis 28 (1991) 401–447. Crossref, Web of Science, Google Scholar
2. K. A. Baker and A. F. Pixley, Polynomial interpolation and the Chinese remainder theorem for algebraic systems, Math. Z. 143 (1975) 165–174. Crossref, Web of Science, Google Scholar
3. C. Bergman, Universal Algebra (Chapman and Hall/CRC, Boca Raton, Florida, 2012). Google Scholar
4. J. Berman, Upper bounds on the sizes of finitely generated algebras, Demonstratio Math. 44 (2011) 447–471. Google Scholar
5. G. Birkhoff, On the structure of abstract algebras, Proc. Camb. Philos. Soc. 29 (1935) 433–454. Crossref, Google Scholar
6. G. Birkhoff, Lattice Theory, 2nd edn. AMS Colloquium Publications (American Mathematical Society, New York, 1948). Google Scholar
7. J. Demetrovics, L. Hannák and L. Rónyai, On the free spectra of maximal clones, C. R. Math. Rep. Acad. Sci. Canada 4 (1982) 363–366. Google Scholar
8. A. L. Foster, The identities of — and unique subdirect factorization within — classes of universal algebras, Math. Z. 62 (1955) 171–188. Crossref, Google Scholar
9. A. L. Foster and A. F. Pixley, Semi-categorical algebras. II, Math. Z. 85 (1964) 169–184. Crossref, Google Scholar
10. R. Freese, E. Kiss and M. Valeriote, Universal algebra calculator, (2011), www.uacalc.org. Google Scholar
11. K. Kaarli and A. F. Pixley, Polynomial Completeness in Algebraic Systems (Chapman and Hall/CRC, Boca Ratan, Florida, 2001). Google Scholar
12. R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Vol. I, (Wadsworth and Brooks/Cole, Belmont, California, 1987). Google Scholar
13. A. F. Pixley, Clusters of algebras: Identities and structure lattices, Ph. D. Dissertation, University of California, Berkeley (1961). Google Scholar
14. A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963) 105–109. Crossref, Google Scholar
15. R. Quackenbush and B. Wolk, Strong representation of congruence lattices, Algebra Universalis 1 (1971) 165–166. Crossref, Google Scholar
16. W. Sierpiński, Sur les fonctions de plusieurs variables, Fund. Math. 33 (1945) 169–173. Crossref, Google Scholar
17. F. M. Sioson, Free-algebraic characterizations of primal and independent algebras, Proc. Amer. Math. Soc. 12 (1961) 435–439. Google Scholar