Korean J. Math. Vol. 30 No. 3 (2022) pp.513-524
DOI: https://doi.org/10.11568/kjm.2022.30.3.513

Epis, dominions and zigzag theorem in commutative groups

Main Article Content

Aftab Hussain Shah
Muneer Nabi
Shabir Ahmad Ahanger

Abstract

In this paper, we introduce the notion of tensor product in groups and prove its existence and uniqueness. Next, we provide the Isbell's zigzag theorem for dominions in commutative groups. We then show that in the category of commutative groups dominions are trivial. This enables us to deduce a well known result epis are surjective in the category of commutative groups.


Article Details

References

[1] Ahanger S. A. and Shah A. H., On zigzag theorem for commutative pomonoids and certain closed and absolutely closed monoids and pomonoids, Beitr ̈age Zur Algebra und Geometrie, 61 (2020) 9–21. Google Scholar

[2] Alam. N. and Khan. N. M., A note on Isbell’s zigzag theorem for commutative semigroups, Algebra Letters, 1 (1) (2012) 22–27. Google Scholar

[3] Higgins, P.M., Epimorphisms and semigroup varieties, PhD. Thesis, Department of Mathemat- ics, Monash University Australia (1983). Google Scholar

[4] Howie, J.M. and Isbell, J.R., Epimorphisms and dominions II, J.Algebra 6, (1967) 7–21 . Google Scholar

[5] Howie, J. M., Fundamentals of Semigroup Theory, Clarendon Press, Oxford (1995). Google Scholar

[6] Isbell, J. R., Epimorphisms and dominions, Proceedings of the Conference on Categorical Algebra, La Jolla, (1965), Lange and Springer, Berlin, (1966), 232–246. Google Scholar

[7] Sohail. N., Zigzag theorem for partially ordered monoids, commun. Algebra, 42 (2014), 2559–2583. Google Scholar

[8] Stenstrom B., Flatness and localization over monoids, Math. Nachr. 48 (1971), 315–334. Google Scholar