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Stability and J-depth of expansions

Published online by Cambridge University Press:  17 April 2009

Jean-Camille Birget
Jean-Camille Birget
Affiliation:
Department of Computer Science, University of Nebraska, Lincoln, NE 68588-0115, United States of America
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Abstract

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In this paper I prove that if a semigroup S is stable then ∧L(S) and ∧R(S) (the Rhodes expansions), and ∧+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function.

The techniques used were originally developed by the author to prove that ∧+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 1 , August 1988 , pp. 41 - 54
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Birget, J.C., ‘Iteration of expansions - Unambiguous semigroups’, J. Pure Appl. Algebra 34 (1984), 1–55.CrossRefGoogle Scholar
[2]Birget, J.C., ‘Arbitrary vs. regular semigroups’, J. Pure Appl. Algebra 34 (1984), 56–115.CrossRefGoogle Scholar
[3]Birget, J.C., ‘Structure of finite semigroups, and generalisations’, (Byleen, K., Jones, P., Pastijn, F. editors), Proceedings of the 1984 Marquette Conference on Semigroups, 1–15.Google Scholar
[4]Birget, J.C., ‘The Synthesis Theorem for finite regular semnigroups and its generalisation’ (to appear).Google Scholar
[5]Birget, J.C., Margolis, S. and Rhodes, J., ‘Finite semigroups whose idempotents from a semnilattice or a band’ (unpublished), see also, Proceedings of the Chico Conference on Semigroups.Google Scholar
[6]Birget, J.C. and Rhodes, J., ‘Almost finite expansions of arbitrary semigroups’, J. Pure Appl. Algebra 32 (1984), 239–287.CrossRefGoogle Scholar
[7]Brown, T.C., ‘An interesting combinatorial method in the theory of locally finite semigroups’, Pacific J. Math. 38 (1971), 285–289.CrossRefGoogle Scholar
[8]Clifford, A.H. and Preston, G., The Algebraic Theory of Semigroups, Vol. 1 (1961), Vol. 2 (1967). (Math. Surveys Monogr., American Mathematical Society, Providence, R.I.)Google Scholar
[9]Eilenberg, S., Automata, Languages, and Machines (Academic Press, New York, 1976).Google Scholar
[10]Lallement, G., Semigroups and Combinatorial Applications (John Wiley, New York, 1979).Google Scholar
[11]Rhodes, J., ‘Infinite iteration of matrix semigroups, Part II’, J. Algebra 100 (1986), 25–137.CrossRefGoogle Scholar
[12]Rhodes, J., Algebraic and topological theory of languages and computation, (Preprint (1983), Center for Pure Applied Math., U. of California, Berkeley).Google Scholar
[13]Rhodes, J., ‘Expansions of semigroups via Ramsey's and Brown's theorem’, (in progress).Google Scholar
[14]Rhodes, J. and Allen, D., ‘Synthesis of classical and modern semigroup theory’, Adv. in Math. 11 (1976), 238–266.CrossRefGoogle Scholar
[15]Rhodes, J. and Tilson, B., Machines, Languages and Semigroups, Chapters 7 and 8 (Arbib, M., editor, Academic Press, New York, 1968).Google Scholar
[16]Straubing, H., ‘The Burnside problem for semigroups of matrices’, in Combinatorics on Works (Cummings, L. editor, Academic Press, New York, 1983).Google Scholar
[17]Tilson, B., ‘On the complexity of finite semigroups’, J. Pure Appl. Algebra 4 (1974).Google Scholar