Isotopic algebras with nonisomorphic congruence lattices.
For algebras \(\mathbf A,\mathbf B,\mathbf C\) of the same type, \(\mathbf A\) is said to be isotopic to \(\mathbf B\) over \(\mathbf C\) (\(\mathbf A\sim_{\mathbf C}\mathbf B\)) if there is an isomorphism \(\varphi\colon\mathbf A\times\mathbf C\to\mathbf B\times\mathbf C\) such that, whenever \(a\in\mathbf A\), \(c\in\mathbf C\), there exists \(b\in\mathbf B\) with \(\varphi(a,c)=(b,c)\). The algebras \(\mathbf A\) and \(\mathbf B\) are called isotopic (\(\mathbf A\sim\mathbf B\)), if \(\mathbf A\sim_{\mathbf C}\mathbf B\) for some algebra \(\mathbf C\). The implication \(\mathbf A\sim\mathbf B\Rightarrow\mathrm{Con\,}\mathbf A\cong\mathrm{Con\,}\mathbf B\) is valid in any congruence modular variety. The author describes a class of algebras, in which this implication fails to hold.
scientific article; zbMATH DE number 6369495