Algebra universalis

, Volume 75, Issue 1, pp 61–74

# Operator properties of congruence permutable varieties with strongly definable principal congruences

Article

## Abstract

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

## References

1. 1.
Bergman G.M.: $${SHPS \neq HSP}$$ for metabelian groups, and related results. Algebra Universalis 26, 267–283 (1989)
2. 2.
Bergman G.M.: Partially ordered sets, and minimal systems of counterexamples. Algebra Universalis 32, 13–30 (1994)
3. 3.
Blok W.J., Pigozzi D.: On the structure of varieties with equationally definable principal congruences I. Algebra Universalis 15, 195–227 (1982)
4. 4.
Burris S.: Remarks on the Fraser-Horn property. Algebra Universalis 23, 19–21 (1986)
5. 5.
Burris S., Sankappanavar H.P.: A Course in universal algebra. Springer, New York (1981)
6. 6.
Fried E., Grätzer G., Quackenbush R.: Uniform congruence schemes. Algebra Universalis 10, 176–188 (1980)
7. 7.
Lawrence J., Tasić B.: $${HSP \neq SHPS}$$ for commutative rings with identity. Proc. Amer. Math. Soc. 134, 943–948 (2006)
8. 8.
Mal’cev I.A.: Algebraic Systems. Akademie, Berlin (1973)
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)
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)
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)
13. 13.
Tasić B.: A note on homomorphic images, subalgebras and various products. Algebra Universalis 52, 431–438 (2004)
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