Operator properties of congruence permutable varieties with strongly definable principal congruences

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 ${\mathcal{K} \subseteq \mathcal{V}}$.