Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate

Question

Given an algebra $\mathbf{A}$, a pair of congruences $ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when $ \alpha \circ \beta =\beta \circ \alpha$, and an algebra $\mathbf{A}$ is called congruence-permutable when each pair of congruences of $\mathbf{A}$ permute. Also, a variety of algebras $\mathcal{V}$ is referred to as congruence-permutable when every algebra in $\mathcal{V}$ is congruence-permutable.

On the other hand, a variety $\mathcal{V}$ is semidegenerate if no nontrivial algebra in $\mathcal{V}$ has one-element subalgebras (equvalently, for any algebra $\mathcal{A}$ in $\mathcal{V}$, the congruence $\nabla_A$ is compact).

I am looking for a reference that proves every congruence-permutable variety is semidegenerate.

Answer 1

I am looking for a reference that prove every congruence-permutable variety is semidegenerate.

It is not true that every congruence permutable variety is semidegenerate. A simple counterexample is the variety of vector spaces over a field $\mathbb F$. (Or the variety of groups, or the variety of $R$-modules for some ring $R$.)

The $\mathbb F$-vector space term operation $x-y+z$ is Maltsev, and the existence of a Maltsev operation is sufficient to guarantee that the variety of $\mathbb F$-spaces is congruence permutable.

The variety of $\mathbb F$-spaces is not semidegenerate, since every vector space has a singleton subalgebra (at the origin).