Algebra universalis

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

Operator properties of congruence permutable varieties with strongly definable principal congruences



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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© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Ted Rogers School of ManagementRyerson UniversityTorontoCANADA

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