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Retraction closure property

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Abstract

We say that an algebra \({\mathcal{A}}\) has the retraction closure property (RCP) if the set of all retractions of \({\mathcal{A}}\) is closed with respect to fundamental operations of \({\mathcal{A}}\) applied pointwise. In this paper we investigate this property, both “locally” (one algebra) and “globally” (in some variety of algebras), especially emphasizing the case of groupoids. We compare the retraction closure property with the endomorphism closure property on both levels and prove that a necessary and sufficient condition for a variety V of algebras to have RCP is that V is a variety of entropic algebras that satisfy the diagonal identities.

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Correspondence to Rozália Madarász.

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Presented by R. Poeschel.

Research supported by the Ministry of Education and Science, Republic of Serbia, Grant No. 174 018.

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Bošnjak, I., Madarász, R. Retraction closure property. Algebra Univers. 69, 279–285 (2013). https://doi.org/10.1007/s00012-013-0229-0

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  • DOI: https://doi.org/10.1007/s00012-013-0229-0

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