Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T14:51:01.019Z Has data issue: false hasContentIssue false
  • English
  • Français

Semigroups whose idempotents form a subsemigroup

Published online by Cambridge University Press:  24 October 2008

J. Almeida ,
J.-E. Pin  and
P. Weil
J. Almeida
Affiliation:
Centro de Matemtica, Universidade do Porto, 4000 Porto, Portugal
J.-E. Pin
Affiliation:
LITP, Universit Paris 6 et CNRS, Tour 5565, 4 place Jussieu, 75252 Paris Cedex 05, France
P. Weil
Affiliation:
LITP, Universit Paris 6 et CNRS, Tour 5565, 4 place Jussieu, 75252 Paris Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every semigroup S in which the idempotents form a subsemigroup has an E-unitary cover with the same property. Furthermore, if S is E-dense or orthodox, then its cover can be chosen with the same property. Then we describe the structure of E-unitary dense semigroups. Our results generalize Fountain's results on semigroups in which the idempotents commute, and are analogous to those of Birget, Margolis and Rhodes, and of Jones and Szendrei on finite E-semigroups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 241 - 253
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Ash, C.. Finite semigroups with commuting idempotents. J. Austral. Math. Soc. Ser. A 43 (1987), 8190.CrossRefGoogle Scholar
2Berstel, J. and Perrin, D.. Theory of Codes (Academic Press, 1985).Google Scholar
3Birget, J.-C., Margolis, S. and Rhodes, J.. Semigroups whose idempotents form a subsemigroup. Bull. Austral. Math. Soc. 41 (1990), 161184.CrossRefGoogle Scholar
4Eilenberg, S.. Automata, Languages and Machines, vol. B (Academic Press, 1976).Google Scholar
5Fountain, J.. E-unitary dense covers of E-dense monoids. Bull. London Math. Soc. 22 (1990), 353358.CrossRefGoogle Scholar
6Jones, P. and Szendrei, M.. Local varieties of completely regular monoids, to appear.Google Scholar
7Kadourek, J. and Szendrei, M.. A new approach in the theory of orthodox semigroups. Semigroup Forum 40 (1990), 257296.CrossRefGoogle Scholar
8Lallement, G.. Semigroups and Combinatorial Applications (Wiley, 1979).Google Scholar
9Margolis, S. and Pin, J.-E.. Inverse semigroups and extensions of groups by semilattices. J. Algebra 110 (1987), 277297.CrossRefGoogle Scholar
10Margolis, S. and Pin, J.-E.. Inverse semigroups and varieties of finite semigroups. J. Algebra 110 (1987), 306323.CrossRefGoogle Scholar
11McAlister, D.. Groups, semilattices and inverse semigroups. Trans. Amer. Math. Soc. 192 (1974), 227244.Google Scholar
12McAlister, D.. Groups, semilattices and inverse semigroups. II. Trans. Amer. Math. Soc. 196 (1974), 251270.CrossRefGoogle Scholar
13McAlister, D. and Reilly, N.. E-unitary covers for inverse semigroups. Pacific J. Math. 68 (1977), 161174.CrossRefGoogle Scholar
14Mitsch, H.. Subdirect products of E-inversive semigroups. J. Austral. Math. Soc. Ser. A 48 (1990), 6878.Google Scholar
15O'Carroll, L.. Embedding theorems for proper inverse semigroups. J. Algebra 42 (1976), 2640.CrossRefGoogle Scholar
16Petrich, M.. Inverse semigroups (Wiley, 1984).Google Scholar
17Pin, J.-E.. Varits de Langages Formels (Masson, 1984)Google Scholar
Pin, J.-E. and Varieties of Formal Languages (North Oxford Academic and Plenum, 1986).CrossRefGoogle Scholar
18Pin, J.-E.. On a conjecture of Rhodes. Semigroup Forum 39 (1989), 115.CrossRefGoogle Scholar
19Pin, J.-E.. Relational morphisms, transductions and operations on languages. In Pin, J.-E. (ed.), Formula Properties of Finite Automata and Applications (ed. Pin, J.-E.), Lecture Notes in Comp. Sci. vol. 386 (Springer-Verlag, 1989) pp. 120137.CrossRefGoogle Scholar
20Reilly, N. and Scheiblich, H.. Congruences on regular semigroups. Pacific J. Math. 23 (1967), 349360.CrossRefGoogle Scholar
21Swift, J. (Gulliver, L.). Travels into several Remote Nations of the World (B. Motte, 1726).Google Scholar
22Szendrei, M.. On a pull-back diagram for orthodox semigroups. Semigroup Forum 20 (1980), 110,CrossRefGoogle Scholar
Szendrei, M.correction 25 (1982), 311324.Google Scholar
23Szendrei, M.. E-unitary R-unipotent semigroups. Semigroup Forum 32 (1985), 8796.CrossRefGoogle Scholar
24Szendrei, M.. E-unitary regular semigroups. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 89102.CrossRefGoogle Scholar
25Szendrei, M.. A generalization of McAlister's P-theorem for E-unitary regular semigroups. Acta Sci. Math. (Szeged) 51 (1987), 229249.Google Scholar
26Takizawa, K.. E-unitary R-unipotent semigroups. Bull. Tokyo Gakugei Univ. (4) 30 (1978), 2133.Google Scholar
27Takizawa, K.. Orthodox semigroups and E-unitary regular semigroups. Bull. Tokyo Gakugei Univ. (4) 31 (1979), 4143.Google Scholar
28Tilson, B.. Categories as algebras: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48 (1987), 3198.CrossRefGoogle Scholar
29Weiss, A. and Thrien, D.. Varieties of finite categories. RAIRO Inform. Thor. Appl. 20 (1986), 357366.CrossRefGoogle Scholar