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A characterization of the Higgins commutator. (English) Zbl 1420.18007
The Higgins commutator [P. J. Higgins, Proc. Lond. Math. Soc. (3) 6, 366–416 (1956; Zbl 0073.01704)] is a universal-algebraic generalization of the commutator of groups to $$\Omega$$-groups. The paper [S. Mantovani and G. Metere, J. Algebra 324, No. 9, 2568–2588 (2010; Zbl 1218.18001)] claims that the Higgins commutator can be defined categorically for a pair of subobjects of each object in an ideal-determined category, while the paper [M. Gran et al., J. Algebra 397, 643–665 (2014; Zbl 1305.18040)] shows that the categorical definition of the Higgins commutator is to be recovered through the weighted subobject commutator. This paper aims at characterizing the Higgins commutator as the largest binary operation $$C$$ defined on all subobjects of each object in an ideal-determined unital category, pursuant to the following conditions:
1.
$$C$$ is order-preserving.
2.
For each pair of subobjects $$H$$ and $$K$$ of an object $$X$$, $$C\left( H,K\right)$$ is always less or equal to the meet of normal closures of $$H$$ and $$K$$.
3.
For every pair of subobjects $$H$$ and $$K$$ of an object $$X$$ and every morphism $$f$$ with domain $$X$$, we have $C\left( f\left( H\right) ,f\left( K\right) \right) =f\left( C\left( H,K\right) \right)$
A similar characterization is to be found in [V. T. Shaumbwa, Theory Appl. Categ. 32, 1588–1600 (2017; Zbl 1397.18010)] for the commutator introduced in [S. A. Huq, Q. J. Math., Oxf. II. Ser. 19, 363–389 (1968; Zbl 0165.03301)] within genuinely categorical setting. The paper [J. Hagemann and C. Herrmann, Arch. Math. 32, 234–245 (1979; Zbl 0419.08001)] gives a characterization of the commutator, introduced in [J. D. H. Smith, Mal’cev varieties. Cham: Springer (1976; Zbl 0344.08002)] and refined categorically in [M. C. Pedicchio, J. Algebra 177, No. 3, 647–657 (1995; Zbl 0843.08004)], on congruence lattices.
##### MSC:
 18B30 Categories of topological spaces and continuous mappings (MSC2010) 18A05 Definitions and generalizations in theory of categories 18B35 Preorders, orders, domains and lattices (viewed as categories) 18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
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##### References:
 [1] Bourn, D., Mal’cev categories and fibration of pointed objects, Appl. Categ. Struct., 4, 307-327, (1996) · Zbl 0856.18004 [2] Gran, M.; Janelidze, G.; Ursini, A., Weighted commutators in semi-abelian categories, J. Algebra, 397, 643-665, (2014) · Zbl 1305.18040 [3] Hagemann, J.; Herrmann, C., A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math., 32, 234-245, (1979) · Zbl 0419.08001 [4] Higgins, PJ, Groups with multiple operators, Proc. Lond. Math. Soc., 6, 366-416, (1956) · Zbl 0073.01704 [5] Huq, SA, Commutator, nilpotency, and solvability in categories, Q. J. Math., 19, 363-389, (1968) · Zbl 0165.03301 [6] Janelidze, G.; Márki, L.; Tholen, W.; Ursini, A., Ideal determined categories, Cah. Topol. Géom. Différ. Catég., 51, 115-125, (2010) · Zbl 1208.18001 [7] Janelidze, Z., The pointed subobject functor, $$3\times 3$$ lemmas, and subtractivity of spans, Theory Appl. Categ., 23, 221-242, (2010) · Zbl 1234.18009 [8] Mantovani, S.; Metere, G., Normalities and commutators, J. Algebra, 324, 2568-2588, (2010) · Zbl 1218.18001 [9] Pedicchio, MC, A categorical approach to commutator theory, J. Algebra, 177, 647-657, (1995) · Zbl 0843.08004 [10] Shaumbwa, VT, A characterization of the Huq commutator, Theory Appl. Categ., 32, 1588-1600, (2017) · Zbl 1397.18010 [11] Smith, J.D.H.: Mal’cev Varieties. Springer Lecture Notes in Mathematics, vol. 554. Springer, Berlin (1976) · Zbl 0344.08002
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