Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T06:43:23.875Z Has data issue: false hasContentIssue false
  • English
  • Français

IDEALS AND FINITENESS CONDITIONS FOR SUBSEMIGROUPS

Published online by Cambridge University Press:  13 August 2013

R. GRAY ,
V. MALTCEV ,
J. D. MITCHELL  and
N. RUŠKUC
R. GRAY
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England e-mail: Robert.D.Gray@uea.ac.uk
V. MALTCEV
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland, United Kingdom e-mail: victor@mcs.st-and.ac.uk, jamesm@mcs.st-and.ac.uk, nik@mcs.st-and.ac.uk
J. D. MITCHELL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland, United Kingdom e-mail: victor@mcs.st-and.ac.uk, jamesm@mcs.st-and.ac.uk, nik@mcs.st-and.ac.uk
N. RUŠKUC
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland, United Kingdom e-mail: victor@mcs.st-and.ac.uk, jamesm@mcs.st-and.ac.uk, nik@mcs.st-and.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, $\mathcal{D}=\mathcal{J}$ and minimal conditions on ideals.

Keywords

20M0520M12
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 65 - 86
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Auinger, K. and Hall, T. E., Representations of semigroups by transformations and the congruence lattice of an eventually regular semigroup. Int. J. Algebra Comput. 6 (1996), 655–685.CrossRefGoogle Scholar
2.Birget, J.-C., The synthesis theorem for finite regular semigroups, and its generalization. J. Pure Appl. Algebra 55 (1988), 1–79.CrossRefGoogle Scholar
3.Book, R. V. and Otto, F., String-rewriting systems, Texts and Monographs in Computer Science (Springer-Verlag, New York, NY, 1993).CrossRefGoogle Scholar
4.Cain, A. J., Gray, R. and Ruškuc, N., Green index in semigroup theory: Generators, presentations, and automatic structures, Semigroup Forum 85 (3) (2012), 448–476. doi:10.1007/s00233-012-9406-2.CrossRefGoogle Scholar
5.Cain, A. J., Robertson, E. F. and Ruškuc, N., Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions, J. Austral. Math. Soc. 84 (2008), 39–61.CrossRefGoogle Scholar
6.Day, D., Expository lectures on topological semigroups, in Algebraic theory of machines, languages and semigroups (Arbib, M. A., Editor) (Academic Press, Atlanta, GA, 1968), 269–296.Google Scholar
7.Easdown, D., Biordered sets of eventually regular semigroups, Proc. Lond. Math. Soc. 49 (1984), 483–503.CrossRefGoogle Scholar
8.Edwards, P. M., Eventually regular semigroups, Bull. Austral. Math. Soc. 28 (1983), 23–38.Google Scholar
9.Elston, G. Z. and Nehaniv, C. L., Holonomy embedding of arbitrary stable semigroups, Int. J. Algebra Comput. 12 (2002), 791–810.Google Scholar
10.Gray, R. and Ruškuc, N., Green index and finiteness conditions for semigroups, J. Algebra 320 (2008), 3145–3164.Google Scholar
11.Higgins, P. M., A class of eventually regular semigroups determined by pseudo-random sets, J. Lond. Math. Soc. 48 (1993), 87–102.CrossRefGoogle Scholar
12.Hoffmann, M., Thomas, R. M. and Ruškuc, N., Automatic semigroups with subsemigroups of finite Rees index, Int. J. Algebra Comput. 12 (2002), 463–476.Google Scholar
13.Howie, J. M., Fundamentals of semigroup theory, LMS Monographs, vol. 7 (Academic Press, London, 1995).Google Scholar
14.Jura, A., Coset enumeration in a finitely presented semigroup, Canad. Math. Bull. 21 (1978), 37–46.CrossRefGoogle Scholar
15.Jura, A., Determining ideals of a given finite index in a finitely presented semigroup, Demonstratio Math. 11 (1978), 813–827.Google Scholar
16.Jura, A., Some remarks on nonexistence of an algorithm for finding all ideals of a given finite index in a finitely presented semigroup, Demonstratio Math. 13 (1980), 573–578.Google Scholar
17.Kambites, M., Presentations for semigroups and semigroupoids, Int. J. Algebra Comput. 15 (2005), 291–308.CrossRefGoogle Scholar
18.Koch, R. J. and Wallace, A. D., Stability in semigroups, Duke Math. J. 24 (1957), 193–195.Google Scholar
19.Lallement, G., Semigroups and combinatorial applications, Pure and Applied Mathematics (John Wiley, New York, NY, 1979).Google Scholar
20.Malheiro, A., On trivializers and subsemigroups, in Semigroups and formal languages (André, J. M.et al., Editors) (World Scientific Publications, Hackensack, NJ, 2007), 188–204.CrossRefGoogle Scholar
21.O'Carroll, L., Counterexamples in stable semigroups, Trans. Amer. Math. Soc. 146 (1969), 377–386.CrossRefGoogle Scholar
22.Rhodes, J. and Steinberg, B., The q-theory of finite semigroups, Springer Monographs in Mathematics (Springer, New York, NY, 2009).Google Scholar
23.Ruškuc, N., On large subsemigroups and finiteness conditions of semigroups, Proc. Lond. Math. Soc. 76 (1998), 383–405.CrossRefGoogle Scholar
24.Ruškuc, N. and Thomas, R. M., Syntactic and Rees indices of subsemigroups, J. Algebra 205 (1998), 435–450.Google Scholar
25.Wallace, A. D., Relative ideals in semigroups II. The relations of Green, Acta Math. Acad. Sci. Hungar. 14 (1963), 137–148.CrossRefGoogle Scholar
26.Wang, J., Finite complete rewriting systems and finite derivation type for small extensions of monoids, J. Algebra 204 (1998), 493–503.CrossRefGoogle Scholar
27.Wong, K. B. and Wong, P. C., On finite complete rewriting systems and large subsemigroups, J. Algebra 345 (2011), 242–256.Google Scholar