Fully invariant and verbal congruence relations
Let \(A\) be an algebra. A congruence \(\theta\) is called verbal if there exists a variety \(V\) such that the factor algebra of \(A\) by \(\theta\) is in \(V\). Every verbal congruence is fully invariant but not conversely. An algebra \(A\) is called verbose if every fully invariant congruence on \(A\) is verbal. A variety is verbose if each algebra in it has this property. The authors prove that for a finite SI-algebra generating congruence-modular (CM in brief) variety its monolith is verbal. Neither finiteness nor CM can be omitted. If \(A\) is a finite algebra in a CM variety and \(\operatorname{Con}A\) is a chain, then \(A\) is verbose. If \(V\) is a locally finite CM minimal variety then \(V\) is verbose. The authors prove several results on verbose varieties which are abelian or semisismple and arithmetical. A number of nice examples is included.
scientific article; zbMATH DE number 6206935
scientific article