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Organizers |
Irreducibility in Equal Length for Finite Semigroups
by
P. Dömösi
Debrecen University, Institute of Mathematics and Informatics, Hungary
By the well-known Krohn-Rhodes theorem (see, for example, [4]) the finite irreducible semigroups are exactly the finite simple groups and the subsemigroups of the monoid with two right-zero elements. It is trivial that all finite cyclic simple groups are irreducible in equal length. Moreover, it is easy to see that all subsemigroups of the monoid with two right-zero elements also have this property. Using the Dénes-Hermann theorem [3], we can also show that all finite non-commutative simple groups are irreducible in equal length. Therefore, the finite irreducible semigroups are irreducible in equal length (and vice versa). (This statement was published by Z. Ésik in [1] and [2].) The only known proof of the Dénes-Hermann theorem uses the Feit-Thompson theorem. Thus Z. Ésik gave a direct proof to show that the finite non-commutative simple groups are irreducible in equal length too. (Actually, this statement is a consequence of his results in [2].) Using an idea of P. P. Pálfy [5], we give another direct, elementary proof of this statement.
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[1] Z. Ésik, An extension of the Krohn-Rhodes decomposition of automata, in: Proc. IMYC`1988, Smolenice, LNCS, 381, Springer, 1989, 66-71.
[2] Z. Ésik, Results on homomorphic realization of automata by \alpha0-products, TCS, 87, 1991, 229-249.
[3] J. Dénes and P. Hermann, On the product of all elements in a finite group, Ann. of Discrete Math., 15, 1982, 107-111.
[4] K. B. Krohn, J. L. Rhodes and B. R. Tilson, The prime decomposition theorem of the algebraic theory of machines, in: M. Arbib, ed., Algebraic Theory of Machines, Languages and Semigroups, Academic Press, New York, 1968.
[5] P. P. Pálfy, On generating systems of non-commutative finite simple groups, personal communication, 1989.
Date received: June 29, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-48.