Skip to main content

On morphisms of commutative monoids


The aim of this work is to study monoid morphisms between commutative monoids. Algorithms to check if a monoid morphism between two finitely generated monoids is injective and/or surjective are given. The structure of the set of monoid morphisms between a monoid and a cancellative monoid is also studied and an algorithm to obtain a system of generators of this set is provided.

This is a preview of subscription content, access via your institution.


  1. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Am. Math. Soc., Providence (1961)

    MATH  Google Scholar 

  2. García García, J.I.: On monoids homomorphisms. In: Actas del Encuentro de Álgebra Computacional y Aplicaciones (EACA’2008), Universidad de Granada (2008)

    Google Scholar 

  3. Geroldinger, A., Halter-Koch, F.: Arithmetical theory of monoid homomorphisms. Semigroup Forum 48, 333–362 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grillet, P.A.: Semigroups. An Introduction to the Structure Theory. Dekker, New York (1995)

    Google Scholar 

  5. Rédei, L.: The Theory of Finitely Generated Commutative Semigroups. Pergamon, Elmsford (1965)

    MATH  Google Scholar 

  6. Rosales, J.C., García-Sánchez, P.A.: Finitely Generated Commutative Monoids. Nova Science Publishers, New York (1999)

    MATH  Google Scholar 

  7. Rosales, J.C., García-Sánchez, P.A., García-García, J.I.: Presentations of finitely generated submonoids of finitely generated commutative monoids. Int. J. Algebra Comput. 12(5), 659–670 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to J. I. García García.

Additional information

Communicated by Jorge Almeida.

J.I. García García was partially supported by MTM2010-15595.

M.A. Moreno Frías was partially supported by MTM2008-06201-C02-02 and FQM-298.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

García García, J.I., Moreno Frías, M.A. On morphisms of commutative monoids. Semigroup Forum 84, 333–341 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Monoid
  • Monoid morphism
  • Congruence
  • Presentation
  • Ordering