On Cohomological Dimension of Ideals of Free Semigroups
by
Boris V. Novikov
Kharkov National University, Kharkov, Ukraine

It was proved in [2] that every cancellative semigroup of cohomological dimension (c.d.) 1 embeds into a free group. The converse is not true [1]. In particular, the problem of studying of subsemigroups of free semigroups with c.d. 1 arises.

Theorem 1 A left ideal of a free semigroup has c.d. 1 iff it is free.

Corollary 2 Every proper two-sided ideal of a free semigroup has c.d.  > 1.

The question for right ideals is open.

Theorem 3 A principal right ideal of a free semigroup has c.d. 1 iff it is free.

The analog of Theorem 1 for right ideals is not true:

Example Let F=<a, b> is a free semigroup. Then R={b, aba}F¹ is not free and c.d. R=1.

Yu. Drozd conjectured that for any S in F either S or S^op (antiisomorphic to S) has c.d. 1. The pair L, L^op, where the principal left ideal L is not free, gives a counter-example to this conjecture.

References

[1]

Novikov, B. V. Partial cohomology of semigroups and its applications , (Russian) Izv. vuzov. Mat. 1988, no. 11, 25-32; translation in Soviet Math. (Iz. VUZ) 32 (1988), no. 11, 38-48.

[2]

Novikov, B. V. Semigroups of cohomological dimension one, J. Algebra 204 (1998), no. 2, 386-393.

Date received: March 9, 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-02.