On clones of polynomials over infinite fields of prime characteristic
by
Alexander Semigrodskikh
Ural State University, Russia

Let K be a field, F_K be the clone of all polynomials in several variables over K, and let L⁰_K be the clone of all linear forms over K. We consider the interval [L⁰_K, F_K] in the lattice of all clones over K.

Theorem 1. If K₁ and K₂ are infinite fields of the same prime characteristic, then the intervals [L⁰_K1, F_K1] and [L⁰_K2, F_K2] are isomorphic.

In [1], the description of [L⁰_K, F_K] is given for every field K of characteristic 0. That description and Theorem 1 imply the following result.

Corollary. If K₁ and K₂ are infinite fields of the same characteristic, then the intervals [L⁰_K1, F_K1] and [L⁰_K2, F_K2] are isomorphic.

The description of [L⁰_K, F_K] for every finite field K is given in [2]. For an infinite field K of a prime characteristic, the interval [L⁰_K, F_K] seems to be very complicated, and the full description of this interval is hardly possible. The following result may confirm this conjecture.

Theorem 2. If K is an infinite field of a prime characteristic, then the interval [L⁰_K, F_K] has the cardinality of the continuum and satisfies no non-trivial lattice identity.

References

1. Semigrodskikh, A. P. On clones of polynomials over infinite fields. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2000, , no. 7, 53-58; English translation in Russian Math. (Iz. VUZ) 44 (2000), no. 7, 50-55.
2. Semigrodskikh, A. P.; Sukhanov, E. V. On closed classes of polynomials over finite fields. (Russian) Diskret. Mat. 9 (1997), no. 4, 50-62; English translation in Discrete Math. Appl. 7 (1997), no. 6, 593-606.

Date received: February 4, 2002

Copyright © 2002 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 # caht-17.