Algebra universalis

, Volume 75, Issue 1, pp 61–74 | Cite as

Operator properties of congruence permutable varieties with strongly definable principal congruences

  • Boža Tasić


In an attempt to describe the partially ordered monoid of operators generated by the operators H (homomorphic images), S (subalgebras), \({P_{\rm f}}\) (filtered products) for the variety \({\mathcal{R}_{\rm c}}\) of commutative rings, several results about congruence permutable varieties have been discovered.

Let us recall that the variety \({\mathcal{R}_{\rm c}}\) is congruence permutable and for any \({\rm {\bf R} \in \mathcal{R}_{\rm c}}\), and \({a, b, c_{1}, d_{1}, . . . c_{k}, d_{k} \in R}\) we have

$$(a, b) \in {\rm C}_{g} ((c_{1}, d_{1}), . . . , (c_{k}, d_{k})) \leftrightarrow \exists e_{1} . . . \exists e_{k} (a - b = \sum_{i=1} ^{k} e_i(c_i - d_i))$$
These two properties are the main reason why \({\mathcal{R}_{\rm c}}\) satisfies \({HP_{\rm f} \leq SP_{\rm f}H}\).

We will actually prove that whenever a congruence permutable variety \({\mathcal{V}}\) has finitely generated congruences definable by a special type of formula, we will have \({HP_{\rm f} (\mathcal{K}) \subseteq SP_{\rm f}HS(\mathcal{K}}\)) for every class \({\mathcal{K} \subseteq \mathcal{V}}\).

Key words and phrases

operators on classes of algebras congruence permutable varieties 

2010 Mathematics Subject Classification

Primary: 06F05 Secondary: 08B05 


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  1. 1.
    Bergman G.M.: \({SHPS \neq HSP}\) for metabelian groups, and related results. Algebra Universalis 26, 267–283 (1989)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bergman G.M.: Partially ordered sets, and minimal systems of counterexamples. Algebra Universalis 32, 13–30 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blok W.J., Pigozzi D.: On the structure of varieties with equationally definable principal congruences I. Algebra Universalis 15, 195–227 (1982)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Burris S.: Remarks on the Fraser-Horn property. Algebra Universalis 23, 19–21 (1986)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Burris S., Sankappanavar H.P.: A Course in universal algebra. Springer, New York (1981)MATHCrossRefGoogle Scholar
  6. 6.
    Fried E., Grätzer G., Quackenbush R.: Uniform congruence schemes. Algebra Universalis 10, 176–188 (1980)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Lawrence J., Tasić B.: \({HSP \neq SHPS}\) for commutative rings with identity. Proc. Amer. Math. Soc. 134, 943–948 (2006)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Mal’cev I.A.: Algebraic Systems. Akademie, Berlin (1973)CrossRefGoogle Scholar
  9. 9.
    McKenzie, R., McNulty, G.F.,Taylor, W.T.: Algebras, Lattices, Varieties, Volume 1. Wadsworth and Brooks/Cole Advanced Books and Software, Monterey (1987)Google Scholar
  10. 10.
    Pigozzi D.: On some operators on classes of algebras. Algebra Universalis 2, 346–353 (1972)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    B. , B. : On the partially ordered monoid generated by the operators \({H, S, P, P_{s}}\) on classes of algebras. J. Algebra 245, 1–19 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Tasić B.: Partially ordered monoids generated by \({H, S, P,}\) and \({H, S, P_{f}}\) are isomorphic. Semigroup Forum 62, 485–490 (2001)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Tasić B.: A note on homomorphic images, subalgebras and various products. Algebra Universalis 52, 431–438 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Tasić, B.: Partially ordered monoids of operators on classes of algebras. PhD thesis, University of Waterloo (2005)Google Scholar
  15. 15.
    Willard, R.: Three lectures on the RS problem. In: Hart, B.T., et al. (eds.) Algebraic Model Theory, pp. 231–254. Kluwer (1997)Google Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Ted Rogers School of ManagementRyerson UniversityTorontoCANADA

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