Abstract
In dealing with monoids, the natural notion of kernel of a monoid morphism \(f:M\rightarrow N\) between two monoids M and N is that of the congruence \(\sim _f\) on M defined, for every \(m,m'\in M\), by \(m\sim _fm'\) if \(f(m)=f(m')\). In this paper, we study kernels and equalizers of monoid morphisms in the categorical sense. We consider the case of the categories of all monoids, commutative monoids, cancellative commutative monoids, reduced Krull monoids, inverse monoids and free monoids. In all these categories, the kernel of \(f:M\rightarrow N\) is simply the embedding of the submonoid \(f^{-1}(1_N)\) into M, but a complete characterization of kernels in these categories is not always trivial, and leads to interesting related notions.
Similar content being viewed by others
References
Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories: the joy of cats”, Wiley, New York, 1990. Reprinted in Reprint Theory Application Category. vol.17 (2006). http://katmat.math.uni-bremen.de/acc/acc
Facchini, A., Halter-Koch, F.: Projective modules and divisor homomorphisms. J. Algebra Appl. 2, 435–449 (2003)
Guba, V.S.: Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems. Mat. Zametki 40, 321–324 (1986). English translation: Math. Notes 40 (1986), 688–690
Hall, T.E.: Free products with amalgamation of inverse semigroups. J. Algebra 34, 375–385 (1975)
Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New Series. The Clarendon Press, Oxford University Press, New York (1995)
Howie, J.M., Isbell, J.R.: Epimorphisms and dominions II. J. Algebra 6, 7–21 (1967)
Isbell, J.R.: Epimorphisms and dominions In: Proceedings of the Conference on Categorical Algebra, La Jolla, California (1965). Springer, New York, pp. 232–246
Post, E.L.: A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52, 264–268 (1946)
Salomaa, A.: Equality sets for homomorphisms on free monoids. Acta Cybernet. 4, 127–139 (1978)
Stephen, J.B.: Amalgamated free products of inverse semigroups. J. Algebra 208, 399–424 (1998)
Acknowledgements
The first author is partially supported by Università di Padova (Progetto ex 60% “Anelli e categorie di moduli”) and Fondazione Cassa di Risparmio di Padova e Rovigo (Progetto di Eccellenza “Algebraic structures and their applications”).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by László Márki.
Rights and permissions
About this article
Cite this article
Facchini, A., Rodaro, E. Equalizers and kernels in categories of monoids. Semigroup Forum 95, 455–474 (2017). https://doi.org/10.1007/s00233-016-9834-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-016-9834-5