The Finite Basis Problem for Semigroups of Orientation Preserving Transformations
by
Mikhail Volkov
Ural State University, Ekaterinburg, Russia

We continue a study of the finite basis problem for pseudovarieties generated by transformation semigroups started in [3], [4]. Here we consider the pseudovariety OP generated by all semigroups of orientation preserving transformations of a finite cycle. The pseudovariety OP has been introduced and studied by Catarino and Higgins [1]. Since OP contains all finite commutative semigroups [1], methods from [3] or [4] designed to handle with aperiodic pseudovarieties do not apply to it. Nevertheless we have managed to solve the finite basis problem for OP by using a construction originated in a recent paper by Higgins and Margolis [2].

Theorem. The pseudovariety OP admits neither a finite pseudoidentity basis nor a finite quasiidentity basis.

References

[1] P. M. Catarino and P. M. Higgins, The pseudovariety generated by all semigroups of orientation-preserving transformations on a finite cycle, submitted.

[2] P. M. Higgins and S. W. Margolis, Finite aperiodic semigroups with commuting idempotents and generalizations, preprint, 1999.

[3] V. B. Repnitskii and M. V. Volkov, The finite basis problem for the pseudovariety O, Proc. Royal Soc. Edinburgh 128A (1998) 661-669.

[4] M. V. Volkov, The finite basis problem for the pseudovariety PO, in J. M. Howie and N. Ruskuc (eds.), Semigroups and Applications, World Scientific, Singapore, 1998, 239-257.

Date received: May 15, 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-31.