Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-22T14:18:25.765Z Has data issue: false hasContentIssue false
  • English
  • Français

Semigroups whose idempotents form a subsemigroup

Published online by Cambridge University Press:  17 April 2009

Jean-Camille Birget ,
Stuart Margolis  and
John Rhodes
Jean-Camille Birget
Affiliation:
Computer Science DepartmentUniversity of NebraskaLincoln, NE 68588USA
Stuart Margolis
Affiliation:
Computer Science DepartmentUniversity of NebraskaLincoln, NE 68588USA
John Rhodes
Affiliation:
Mathematics DepartmentUniversity of CaliforniaBerkeley, CA 94720USA
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.

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:

(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.

(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.

(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.

For these three classes of semigroups, type-II is equal to type-II construct.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 2 , April 1990 , pp. 161 - 184
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Ash, C.J., ‘Finite idempotent-commuting semigroups’, in Semigroups and their applications, Editors Goberstein, S. and Higgins, P., pp. 13–25 (Reidel, Dordrecht, 1986).Google Scholar
[2]Ash, C.J., ‘Finite semigroups with commuting idempotents’, J. Austra. Math. Soc. 43 (1987), 81–90.CrossRefGoogle Scholar
[3]Birget, J.C., Margolis, S. and Rhodes, J., ‘Finite semigroups whose idempotents commute or form a subsemigroup’, in Semigroups and their applications, Editors Goberstein, S. and Higgins, P., pp. 25–36 (Reidel, Dordrecht, 1986).Google Scholar
[4]Birget, J.C., ‘Iteration of expansions, unambiguous semigroups’, J. Pure Appl. Algebra 34 (1984), 1–55.CrossRefGoogle Scholar
[5]Birget, J.C. and Rhodes, J., ‘Almost finite expansions of arbitrary semigroups’, J. Pure Appl. Algebra 32 (1984), 239–287.CrossRefGoogle Scholar
[6]Brown, T.C., ‘An interesting combinatorial method in the theory of locally finite semigroups’, Pacific J. Math. 36 (1971), 285–289.CrossRefGoogle Scholar
[7]Eilenberg, S., Automata, Languages and Machines Vol. B (Academic Press, New York, 1976).Google Scholar
[8]Gerhard, J.A., ‘The lattice of equational classes of idempotent semigroups’, J. Algebra 15 (1970), 195–224.CrossRefGoogle Scholar
[9]Knast, R., ‘A semigroup characterization of dot-depth one languages’, RAIRO Inform. Théor. Appl. 17 (1983), 321–330.CrossRefGoogle Scholar
[10]Krohn, K., Rhodes, J. and Tilson, B., ‘Lectures on finite semigroups’, Chapters 1, 5–9, in Algebraic Theory of Machines, Languages and Semigroups, Editor Arbib, M. (Academic Press, New York, 1968).Google Scholar
[11]Karnofsky, J. and Rhodes, J., ‘Decidability of complexity one-half for finite semigroups’ 24: Semigroup Forum, pp. 55–66.CrossRefGoogle Scholar
[12]Lallement, G., Semigroups and combinatorial applications (Wiley, New York, 1979).Google Scholar
[13]Margolis, S.W., ‘Problem 14’: Proceedings of the Nebraska conference on semigroups, Editor Meakin, p. 14.Google Scholar
[14]Margolis, S.W. and Pin, J.E., ‘Inverse semigroups and extensions of groups by semilattices’, J. Algebra 110 (1987), 277–297.CrossRefGoogle Scholar
[15]Margolis, S.W. and Pin, J.E., ‘Expansions, free inverse semigroups, and Schützenberger product’, J. Algebra 110 (1987), 298–305.CrossRefGoogle Scholar
[16]Margolic, S.W. and Pin, J.E., ‘Inverse semigroups and varieties of finite semigroups’, J. Algebra 110 (1987), 306–323.CrossRefGoogle Scholar
[17]MacAlister, D.B., ‘Regular semigroups, fundamental semigroups and groups’, Ser. A, J. Austral. Math. Soc. 29 (1980), 475–503.CrossRefGoogle Scholar
[18]Pin, J.E., Varieties of formal languages (Plenum Press, New York, 1986).CrossRefGoogle Scholar
[19]Rhodes, J., ‘New techniques in global semigroup theory’, in Semigroups and their applications, Editors Goberstein, S. and Higgins, P., pp. 169–182 (Reidel, Dordrecht, 1986).Google Scholar
[20]Rhodes, J., ‘Global structure theorems for arbitrary semigroups’: Proc. of the 1984 Marquette Conf. on Semigroups. Editors Byleen, K., Jones, P. and Pastijn, F..Google Scholar
[21]Rhodes, J. and Tilson, B., ‘Improved lower bounds for the complexity of finite semigroups’, J. Pure Appl. Algebra 2 (1972), 13–71.CrossRefGoogle Scholar
[22]Simon, I., (Simon's lemma in [7]).Google Scholar
[23]Tilson, B., ‘Type II Redux’, in Semigroups and their applications, Editors Goberstein, S. and Higgins, P., pp. 201–206 (Reidel, Dordrecht, 1986).Google Scholar
[24]Tilson, B., ‘Categories as algebra’, J. Pure Appl. Algebra 48 (1987), 83–198.CrossRefGoogle Scholar
[25]Thérien, D., ‘On the equation xt = xt+g in categories’, Semigroup Forum 37 (1988), 265–272.CrossRefGoogle Scholar
[26]Thérien, D. and Weiss, A., ‘Varieties of finite categories’, RAIRO Inform Théor. Appl. 20 (1986).Google Scholar
[27]Thérien, D. and Weiss, A., ‘Graph congruences and wreath product’, J. Pure Appl. Algebra 36 (1985), 205–215.CrossRefGoogle Scholar