Open Journal Systems


Nwawuru Francis

Article ID: 671
Vol 1, Issue 1, 2018, Article identifier:

VIEWS - 407 (Abstract) 364 (PDF)


Let and  be two free semigroups. We define external direct product of two free semigroups as an ordered pair of words such  that and .We investigate the presentations of external direct product of free semigroups, state and prove under some conditions that the external direct product of two finitely generated free semigroups is finitely generated, also the external direct product of two finitely presented free semigroups is finitely presented. 


free semigroup

Full Text:


Included Database


Alan, J. C. (2015). Nine chapters on the semigroup art. California: Porto & Lisbon.

Campbell, C. M., Robertson, E. F. & Thomas, R. M. (1998). Presentations for subsemigroups -applications to ideals of semigroups. Journal of pure and applied algebra, 124, 47-64. Doi: 10.1016/SOO…

Catarina Carvalho, Presentations of Semigroups and Inverse Semigroups’, M.Sc. Dissertation, University of St. Andrews, February 26, 2003

Clifford, A. H., & Preston, G.B. (1961). The algebraic theorem of semigroups, American Mathematical Society Mathematical Surveys N0. 7, Vol I.

Harju, T. P (1996). Lecture Notes on Semigroups. FIN-20014 TURKU, Finland.

Howie, J. M. (1995). Fundamentals of Semigroup theory. Oxford: Oxford University Press.

I. Araujo and N. Ruskuc, On finite presentability of direct products of semigroups, Algebra Colloq. 7 (2000), 83–91.

Robert D. Gray, The minimum number of generators of a finite Semigroup, Springer 2013, Vol. 89, Issue 1, pp. 135-154

Wiegold, J., & Clark, L. P. (n.p.). Introduction to Semigroup and Monoid. Retrieved on 25 February, 2016 from

Robertson E.F., Ruskuc N. and J. Wiegold, Generators and relations of direct products of

Semigroups, Trans. Amer. Math. Soc. 350 (1998), 2665–2685.

Ruskuc, N., ‘Semigroup Presentations’, Ph.D. Thesis, University of St. Andrews,

Semigroups (n.d.). Retrieved on 25th February, 2016 from…/UST-chapter 0

Wikipedia the free encyclopedia. (n.d.) Free moniod. Retrieved on 25 February, 2016 from

Rida-E Z., “Decomposition of semigroups into semidirect and Zappa-Szép products”, A Ph. D thesis, University of York, August 2014.

Gray R., and Ruskuc N., ‘Generators and Relations for Subsemigroups via Boundaries in Cayley Graphs’, Journal of pure and Applied Algebra, Vol 215, no. 11, pp.2761-2779.,106/j.japaa, 2011. 03.017

Sri G., and Abd. Ghafur Bin Ahmad, ‘J. Indones. Math. Soc. (MIHMI) Vol. 13, No. 2 (2007), pp. 209(214).

(407 Abstract Views, 364 PDF Downloads)


  • There are currently no refbacks.

Copyright (c) 2018 Nwawuru Francis

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.