Groups of Permutation Identities for Some Classes of Semigroups
by
Marin Gutan
Département de Mathématiques et Informatique, Université Blaise Pascal, Complexe scientifique des Cézeaux, Aubière Cedex, France

If \sigma is a permutation of degree n >= 2 , a semigroup S is said to be \sigma-permutable if x₁... x_n = x_\sigma(1)... x_\sigma(n), for all n-tuple x₁, ... , x_n in S. Let G_n(S) = {\sigma in \Sigma_n | S  is  \sigma-permutable}. Notice that G_n(S) is a subgroup of S_n, called the group of permutation identities of degree n of the semigroup S. Some properties of these groups have been established in [1], [2] and [3], particularly an open problem on (m, n)- commutative semigroups, raised up by S. Lajos, has been solved in [2] and [3] (S is (m, n)-commutative if xy = yx, for all x in S^m and y in Sⁿ).

The aim of my talk is to describe the sequence (G_n(S))_{n >= 2} when S belongs to some remarquable classes of semigroups.

[1] M. Gutan, Sur les semi-groupes satisfaisant des identités permutationnelles, C. R. Acad. Sci. Paris, Série I (1994), 5-10.

[2] M. Gutan, A problem on semigroups satisfying permutation identities, Semigroup Forum, 53 (1996), 173-183.

[3] A. Kisielewicz, Permutability class of semigroups, J. Algebra 226 (2000), 295-310.

Date received: May 31, 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-39.