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Representability types of varieties and Mal’tsev strict conditions. (English. Russian original) Zbl 0849.08006

Sib. Math. J. 35, No. 3, 614-642 (1994); translation from Sib. Mat. Zh. 35, No. 3, 683-695 (1994).
We develop a model-theoretic approach to the study of Mal’tsev strict conditions (SCs). We prove that every locally independent set of SCs is independent and therefore the implication relation on SCs possesses the compactness property. We find a necessary condition for the existence of an independent generating set (basis) for a Mal’tsev theory in terms of coverability and \(\vee\)-irreducibility. It permits us to establish that every theory \({\mathcal T} \in {\mathbf L}^{sc}\), \({\mathcal T} \neq 0\), with a basis \(\Sigma\) includes \(|\Sigma|\) maximal subtheories \({\mathcal T}_1, {\mathcal T}_2,\dots\) such that, for every proper subset \(\Delta \subset \Sigma\), the subtheory \(\text{SC} (\Delta)\) generated by \(\Delta\) is included in \({\mathcal T}_k\) for some \(k\).

MSC:

08B05 Equational logic, Mal’tsev conditions
03C05 Equational classes, universal algebra in model theory
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References:

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