Abstract
We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.
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N. Martins-Ferreira was supported by IPLeiria/ESTG-CDRSP and Fundação para a Ciência e a Tecnologia (grant number SFRH/BPD/4321/2008).
T. Van der Linden was supported by Centro de Matemática da Universidade de Coimbra and by Fundação para a Ciência e a Tecnologia (grant number SFRH/BPD/38797/2007). He wishes to thank the Instituto Politécnico for its kind hospitality during his stay in Leiria.
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Martins-Ferreira, N., Van der Linden, T. A Note on the “Smith is Huq” Condition. Appl Categor Struct 20, 175–187 (2012). https://doi.org/10.1007/s10485-010-9231-2
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DOI: https://doi.org/10.1007/s10485-010-9231-2
Keywords
- Commutator
- Internal reflexive graph
- Star-multiplication
- Groupoid
- Protomodular category
- Semi-abelian category