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Linear extensions and nilpotence of Maltsev theories. (English) Zbl 1081.18008

The authors introduce a notion of linear extension of algebraic theories, which can be thought of as analogous to the notion of an extension of groups with abelian kernel. It turns out that many interesting properties of algebraic theories are preserved under linear extensions: in particular, if a theory contains a Mal’tsev operation, then so do all its linear extensions. The authors show that this notion of linear extension of Mal’tsev theories agrees well with the older notion of central extension arising from the intrinsic commutator calculus of Mal’tsev varieties. They also give a description of the ‘abelian’ Mal’tsev theories from which nilpotent theories may be built up by iterated ‘untwisted’ linear extensions.

MSC:

18C10 Theories (e.g., algebraic theories), structure, and semantics
08B05 Equational logic, Mal’tsev conditions
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