Variations of the shifting lemma and Goursat categories. (English) Zbl 1472.08009
Summary: We prove that Mal’tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences \(R\), \(S\) and \(T\), and characterizes congruence modular varieties. We first show that a regular category \(\mathbb{C}\) is a Mal’tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in \(\mathbb{C}\). Moreover, we prove that a regular category \(\mathbb{C}\) is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation \(S\) and reflexive and positive relations \(R\) and \(T\) in \(\mathbb{C}\). In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras.
MSC:
| 08C05 | Categories of algebras |
| 08B05 | Equational logic, Mal’tsev conditions |
| 08A30 | Subalgebras, congruence relations |
| 08B10 | Congruence modularity, congruence distributivity |
| 18C05 | Equational categories |
| 18B99 | Special categories |
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
Mal’tsev categories; Goursat categories; shifting lemma; congruence modular varieties; 3-permutable varietiesReferences:
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