Varieties of shells
An algebra \((A;\cdot ,+,0,1)\) of type \((2,2\), \(0,0)\) is a shell if it satisfies the identities \(x\cdot 0=0 \cdot x=0\), \(x\cdot 1=1\cdot x=x\) and \(x+0=0+x=x.\) The author proves: Theorem. Let \({\mathcal V}\) be a Mal'tsev variety. The following conditions are equivalent: (a) \(A\times A\) is a compact congruence in \(\text{Con} A\) for each \(A\in{\mathcal V}\); (b) no SI member of \(\mathcal V\) has a trivial subalgebra; (c) \(\mathcal V\) is a variety of shells. Let us mention that (a)\(\Longleftrightarrow \)(b) was proven by \textit{J. Kollár} in 1979 without the assumption that \(\mathcal V\) is a Mal'tsev variety. Moreover, if \({\mathcal V}\) is a Mal'tsev variety with two nullary operations, (b)\(\Longleftrightarrow \)(c) can be easily derived from a result of \textit{B. Csákány} [Colloq. Math. 35, 201-203 (1976; Zbl 0331.08002)].
scientific article; zbMATH DE number 1226562
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