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GENERAL THEOREMS ON AUTOMORPHISMS OF SEMIGROUPS AND THEIR APPLICATIONS

Part of: Semigroups

Published online by Cambridge University Press:  01 August 2009

JOÃO ARAÚJO  and
JANUSZ KONIECZNY
JOÃO ARAÚJO
Affiliation:
Universidade Aberta, R. Escola Politécnica, 147, 1269-001 Lisboa, Portugal Centro de Álgebra, Universidade de Lisboa, 1649-003 Lisboa, Portugal (email: [email protected])
JANUSZ KONIECZNY*
Affiliation:
Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We introduce the notion of a strong representation of a semigroup in the monoid of endomorphisms of any mathematical structure, and use this concept to provide a theoretical description of the automorphism group of any semigroup. As an application of our general theorems, we extend to semigroups a well-known result concerning automorphisms of groups, and we determine the automorphism groups of certain transformation semigroups and of the fundamental inverse semigroups.

Keywords

automorphismsemigroupmonoid of endomorphismsrepresentation

MSC classification

Secondary: 20M15: Mappings of semigroups 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 1 - 17
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

The first author was partially supported by FCT and FEDER, Project POCTI-ISFL-1-143 of Centro de Algebra da Universidade de Lisboa, and by FCT and PIDDAC through the project PTDC/MAT/69514/2006.

References

[1]Araújo, J., ‘Aspects of the endomorphism monoids of independence algebras’, PhD Thesis, University of York, 2000.Google Scholar
[2]Araújo, J., Dobson, E. and Konieczny, J., ‘Automorphisms of endomorphism semigroups of reflexive digraphs’, Math. Nachr. to appear.Google Scholar
[3]Araújo, J. and Konieczny, J., ‘Automorphism groups of centralizers of idempotents’, J. Algebra 269 (2003), 227–239.CrossRefGoogle Scholar
[4]Araújo, J. and Konieczny, J., ‘Automorphisms of endomorphism monoids of relatively free bands’, Proc. Edinburgh Math. Soc. (2) 50 (2007), 1–21.CrossRefGoogle Scholar
[5]Araújo, J. and Konieczny, J., ‘Automorphisms of endomorphism monoids of 1-simple free algebras’, Comm. Algebra 37 (2009), 83–94.CrossRefGoogle Scholar
[6]Dixon, J. D. and Mortimer, B., Permutation Groups, Graduate Texts in Mathematics, 163 (Springer, New York, 1996).CrossRefGoogle Scholar
[7]Formanek, E., ‘A question of B. Plotkin about the semigroup of endomorphisms of a free group’, Proc. Amer. Math. Soc. 130 (2002), 935–937.CrossRefGoogle Scholar
[8]Gluskǐn, L. M., ‘Semigroups and rings of endomorphisms of linear spaces I’, Amer. Math. Soc. Transl. 45 (1965), 105–137.Google Scholar
[9]Herrlich, H. and Strecker, G. E., Category Theory: An Introduction (Allyn and Bacon, Boston, MA, 1973).Google Scholar
[10]Howie, J. M., Fundamentals of Semigroup Theory (Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[11]Levi, I., ‘Automorphisms of normal transformation semigroups’, Proc. Edinburgh Math. Soc. (2) 28 (1985), 185–205.CrossRefGoogle Scholar
[12]Levi, I., ‘Automorphisms of normal partial transformation semigroups’, Glasgow Math. J. 29 (1987), 149–157.CrossRefGoogle Scholar
[13]Levi, I. and Seif, S., ‘Finite normal semigroups’, Semigroup Forum 57 (1998), 69–74.CrossRefGoogle Scholar
[14]Liber, A. E., ‘On symmetric generalized groups’, Mat. Sbornik N.S. 33 (1953), 531–544; (in Russian).Google Scholar
[15]Magill, K. D., ‘Semigroup structures for families of functions, I. Some homomorphism theorems’, J. Aust. Math. Soc. 7 (1967), 81–94.CrossRefGoogle Scholar
[16]Mal′cev, A. I., ‘Symmetric groupoids’, Mat. Sbornik N.S. 31 (1952), 136–151; (in Russian).Google Scholar
[17]Mashevitzky, G. and Schein, B. M., ‘Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup’, Proc. Amer. Math. Soc. 131 (2003), 1655–1660.CrossRefGoogle Scholar
[18]Munn, W. D., ‘Uniform semilattices and bisimple inverse semigroups’, Quart. J. Math. Oxford Ser. (2) 17 (1966), 151–159.CrossRefGoogle Scholar
[19]Rips, E., ‘Preface [Special issue on papers from the Conference in Honor of the 80th Birthday of Professor Boris I. Plotkin]’, Internat. J. Algebra Comput. 17 (2007), v–vii.Google Scholar
[20]Schein, B. M., ‘Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups’, Fund. Math. 68 (1970), 31–50.CrossRefGoogle Scholar
[21]Schreier, I., ‘Über Abbildungen einer abstrakten Menge Auf ihre Teilmengen’, Fund. Math. 28 (1936), 261–264.CrossRefGoogle Scholar
[22]Scott, W. R., Group Theory (Prentice Hall, Englewood Cliffs, NJ, 1964).Google Scholar
[23]Sullivan, R. P., ‘Automorphisms of transformation semigroups’, J. Aust. Math. Soc. 20 (1975), 77–84.CrossRefGoogle Scholar
[24]Šutov, È. G., ‘Homomorphisms of the semigroup of all partial transformations’, Izv. Vysš. Učebn. Zaved. Mat. 3 (1961), 177–184; (in Russian).Google Scholar