On the ternary commutator. I: Exact Mal’tsev categories.

*(English)*Zbl 1467.18019A. Bulatov [Contrib. Gen. Algebra 13, 41–54 (2001; Zbl 0986.08003)] introduced a higher-order \(n\)-ary commutator operator of congruences in Mal’tsev algebras, which extends the binary smith commutator [J. D. H. Smith, Mal’cev varieties. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0344.08002)] and is based on a generalization of the term condition. E. Aichinger and N. Mudrinski [Algebra Univers. 63, No. 4, 367–403 (2010; Zbl 1206.08003)] have developed further the higher-order commutator theory in the context of Mal’tsev varieties, where they found analogues for the binary Smith commutator such as monotonicity, stability with respect to joins, stability with respect to restriction, and so on. J. Opršal [Algebra Univers. 76, No. 3, 367–383 (2016; Zbl 1357.08002)] introduced a relational description of the higher-order commutator in Mal’tsev varieties, studying the connection between the term condition and a certain \(n\)-fold relation, called the algebra of \(2^{n}\)-matrices, which can be seen as a higher-order version of the double relation \(\Delta\left( R,S\right) \).

The theory of higher commutators has recently been extended beyond Mal’tsev varieties. A. Moorhead [J. Algebra 513, 133–158 (2018; Zbl 1414.08003)] used the term condition as a basis for higher-order commutator theory in congruence modular varieties, while he [“Some notes on the ternary modular commutator”, Preprint, arXiv:1808.01407] introduced a concept of higher centrality based on matrix constructions leading to a connection with the term condition in congruence modular varieties and a characterization of the ternary commutator in terms of a three-fold relation \(\Delta\left( R,S,T\right) \). A. Wires [Algebra Univers. 80, No. 1, Paper No. 1, 37 p. (2019; Zbl 1439.08007)] has developed further higher commutator properties outside congruence modular varieties.

The principal objective in this paper is to give a categorical description of the Bulatov commutator in the context of exact Mal’tsev categories, extending [M. C. Pedicchio, J. Algebra 177, No. 3, 647–657 (1995; Zbl 0843.08004); D. Bourn and M. Gran, Algebra Univers. 48, No. 3, 309–331 (2002; Zbl 1061.18006); F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1061.18001)]. It is shown to have many of the convenient properties of the universal-algebraic counterparts. After Moorhead’s work, the authors introduce a construction of a three-fold equivalence relation \(\Delta\left( R,S,T\right) \) based on Pedicchio’s \(\Delta\left( R,S\right) \), which in turn develops into \(3\)-fold \(\Delta\)-equivalence relations, the concept of a \(3\) -dimensional connector, and a ternary Bulatov commutator \(\left[ R,S,T\right] ^{\mathrm{B}}\). It is quite clear that the results in this paper can in principle be extended to higher orders.

The authors’ future work goes as follows.

Some open problems remain.

The theory of higher commutators has recently been extended beyond Mal’tsev varieties. A. Moorhead [J. Algebra 513, 133–158 (2018; Zbl 1414.08003)] used the term condition as a basis for higher-order commutator theory in congruence modular varieties, while he [“Some notes on the ternary modular commutator”, Preprint, arXiv:1808.01407] introduced a concept of higher centrality based on matrix constructions leading to a connection with the term condition in congruence modular varieties and a characterization of the ternary commutator in terms of a three-fold relation \(\Delta\left( R,S,T\right) \). A. Wires [Algebra Univers. 80, No. 1, Paper No. 1, 37 p. (2019; Zbl 1439.08007)] has developed further higher commutator properties outside congruence modular varieties.

The principal objective in this paper is to give a categorical description of the Bulatov commutator in the context of exact Mal’tsev categories, extending [M. C. Pedicchio, J. Algebra 177, No. 3, 647–657 (1995; Zbl 0843.08004); D. Bourn and M. Gran, Algebra Univers. 48, No. 3, 309–331 (2002; Zbl 1061.18006); F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1061.18001)]. It is shown to have many of the convenient properties of the universal-algebraic counterparts. After Moorhead’s work, the authors introduce a construction of a three-fold equivalence relation \(\Delta\left( R,S,T\right) \) based on Pedicchio’s \(\Delta\left( R,S\right) \), which in turn develops into \(3\)-fold \(\Delta\)-equivalence relations, the concept of a \(3\) -dimensional connector, and a ternary Bulatov commutator \(\left[ R,S,T\right] ^{\mathrm{B}}\). It is quite clear that the results in this paper can in principle be extended to higher orders.

The authors’ future work goes as follows.

- ●
- In a forthcoming article, the authors restrict the context to algebraically coherent [A. S. Cigoli et al., Theory Appl. Categ. 30, 1864–1905 (2015; Zbl 1366.18005)], semi-abelian [G. Janelidze et al., J. Pure Appl. Algebra 168, No. 2–3, 367–386 (2002; Zbl 0993.18008)] categories, establishing that the commutator introduced in this paper corresponds to the ternary Higgins commutator in [M. Hartl and T. Van der Linden, Adv. Math. 232, No. 1, 571–607 (2013; Zbl 1258.18007)].
- ●
- The authors are currently struggling to generalize the binary Smith-Pedicchio commutator to a higher-order version, which may be exploited to characterize higher central extensions in the sense of [T. Everaert et al., Adv. Math. 217, No. 5, 2231–2267 (2008; Zbl 1140.18012)].

Some open problems remain.

- ●
- The availability of the commutator in congruence modular varieties [A. Moorhead, J. Algebra 513, 133–158 (2018; Zbl 1414.08003)] suggests that the categorical counterpart might be extended beyond the exact Mal’tsev context.
- ●
- The relationship between \(2\)-nilpotency defined in terms of the commutator considerd in this paper and the \(2\)-folded objects in [C. Berger and D. Bourn, J. Homotopy Relat. Struct. 12, No. 4, 765–835 (2017; Zbl 1397.18021)] remains open, being related to the main question in [A. Moorhead, Trans. Am. Math. Soc. 374, No. 2, 1229–1276 (2021; Zbl 1475.08001)] concerning the relationship between so-called supernilpotency (defined in terms of higher-order commutators) and nilpotency (defined in terms of binary commutators).

Reviewer: Hirokazu Nishimura (Tsukuba)

##### MSC:

18E13 | Protomodular categories, semi-abelian categories, Mal’tsev categories |

##### Citations:

Zbl 0986.08003; Zbl 0344.08002; Zbl 1206.08003; Zbl 1357.08002; Zbl 1414.08003; Zbl 1439.08007; Zbl 0843.08004; Zbl 1061.18006; Zbl 1061.18001; Zbl 1366.18005; Zbl 0993.18008; Zbl 1258.18007; Zbl 1140.18012; Zbl 1397.18021; Zbl 1475.08001
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\textit{C. S. Simeu} and \textit{T. Van der Linden}, Theory Appl. Categ. 36, 379--422 (2021; Zbl 1467.18019)

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##### References:

[1] | E. Aichinger and N. Mudrinski, Some applications of higher commutators in Mal’cev algebras, Algebra Universalis 63 (2010), 367403. · Zbl 1206.08003 |

[2] | C. Berger and D. Bourn, Central reections and nilpotency in exact Mal’tsev categories, J. Homotopy Relat. Struct. 12 (2017), 765835. · Zbl 1397.18021 |

[3] | F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004. · Zbl 1061.18001 |

[4] | D. Bourn, The denormalized33lemma, J. Pure Appl. Algebra 177 (2003), 113129. |

[5] | D. Bourn, Commutator theory in strongly protomodular categories, Theory Appl. Categ. 13 (2004), no. 2, 2740. · Zbl 1068.18006 |

[6] | D. Bourn, Fibration of points and congruence modularity, Algebra Universalis 52 (2004), 403429. · Zbl 1086.08003 |

[7] | D. Bourn and M. Gran, Centrality and connectors in Maltsev categories, Algebra Universalis 48 (2002), 309331. · Zbl 1061.18006 |

[8] | D. Bourn and M. Gran, Centrality and normality in protomodular categories, Theory Appl. Categ. 9 (2002), no. 8, 151165. · Zbl 1004.18004 |

[9] | D. Bourn and M. Gran, Categorical aspects of modularity, Galois Theory, Hopf Algebras, and Semiabelian Categories (G. Janelidze, B. Pareigis, and W. Tholen, eds.), Fields Inst. Commun., vol. 43, Amer. Math. Soc., 2004. · Zbl 1081.08011 |

[10] | D. Bourn and M. Gran, Normal sections and direct product decompositions, Comm. Algebra 32 (2004), no. 10, 38253842. · Zbl 1062.18003 |

[11] | A. Bulatov, On the number of nite Mal’tsev algebras, Contribution to general algebra 13, Proceedings of the Dresden Conference 2000 and the Summer School 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 4154. |

[12] | A. Carboni, G. M. Kelly, and M. C. Pedicchio, Some remarks on Maltsev and Goursat categories, Appl. Categ. Structures 1 (1993), 385421. · Zbl 0799.18002 |

[13] | A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, Algebraically coherent categories, Theory Appl. Categ. 30 (2015), no. 54, 18641905. · Zbl 1366.18005 |

[14] | T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf formulae for homology via Galois Theory, Adv. Math. 217 (2008), no. 5, 22312267. · Zbl 1140.18012 |

[15] | T. Everaert and T. Van der Linden, A note on double central extensions in exact Mal’tsev categories, Cah. Topol. Géom. Dier. Catég. LI (2010), 143153. · Zbl 1215.18013 |

[16] | R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc. Lecture Note Ser., vol. 125, Cambridge Univ. Press, 1987. · Zbl 0636.08001 |

[17] | M. Gran, Central extensions and internal groupoids in Maltsev categories, J. Pure Appl. Algebra 155 (2001), 139166. · Zbl 0974.18007 |

[18] | M. Gran and D. Rodelo, The cuboid lemma and Mal’tsev categories, Appl. Categ. Structures 22 (2014), 805816. · Zbl 1305.18009 |

[19] | M. Hartl and T. Van der Linden, The ternary commutator obstruction for internal crossed modules, Adv. Math. 232 (2013), no. 1, 571607. · Zbl 1258.18007 |

[20] | P. J. Higgins, Groups with multiple operators, Proc. Lond. Math. Soc. (3) 6 (1956), no. 3, 366416. · Zbl 0073.01704 |

[21] | G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 23, 367386. · Zbl 0993.18008 |

[22] | S. Mantovani and G. Metere, Normalities and commutators, J. Algebra 324 (2010), no. 9, 25682588. · Zbl 1218.18001 |

[23] | A. Moorhead, Higher commutator theory for congruence modular varieties, J. Algebra 513 (2018), 133158. · Zbl 1414.08003 |

[24] | A. Moorhead, Some notes on the ternary modular commutator, preprint arXiv:1808.01407, 2018. · Zbl 1414.08003 |

[25] | A. Moorhead, Supernilpotent Taylor algebras are nilpotent, Trans. Amer. Math. Soc. 374 (2021), 12291276. · Zbl 1475.08001 |

[26] | J. Opr²al, A relational description of higher commutators in Mal’cev varieties, Algebra Universalis 76 (2014), 367383. |

[27] | M. C. Pedicchio, A categorical approach to commutator theory, J. Algebra 177 (1995), 647657. · Zbl 0843.08004 |

[28] | M. C. Pedicchio, Arithmetical categories and commutator theory, Appl. Categ. Structures 4 (1996), no. 23, 297305. · Zbl 0939.18005 |

[29] | D. Rodelo and T. Van der Linden, Higher central extensions and cohomology, Adv. Math. 287 (2016), 31108. · Zbl 1333.18022 |

[30] | J. D. H. Smith, Mal’cev varieties, Lecture Notes in Math., vol. 554, Springer, 1976. · Zbl 0344.08002 |

[31] | A. Wires, On supernilpotent algebras, Algebra Universalis 80 (2019), no. 1. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, B1348 Louvain-la-Neuve, Belgium Email: cyrille.simeu@uclouvain.be tim.vanderlinden@uclouvain.be · Zbl 1439.08007 |

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