Maximal T-spaces of a free associative algebra

TL;DR

This paper investigates the structure of T-spaces in free associative algebras over fields, revealing a maximum element when the field is infinite and infinitely many maximal elements when the field is finite.

Contribution

It establishes the existence of a maximum T-space in infinite fields and multiple maximal T-spaces in finite fields, extending the understanding of T-space lattices in free associative algebras.

Findings

Maximum T-space exists for infinite fields.

Infinitely many maximal T-spaces for finite fields.

Results transfer from single-element to general case via bijection.

Abstract

We study the lattice of T-spaces of a free associative k-algebra over a nonempty set. It is shown that when the field k is infinite, then the lattice has a maximum element, and that maximum element is in fact a T-ideal. In striking contrast, it is then proven that when the field k is finite, the lattice of T-spaces has infinitely many maximal elements (of which exactly two are T-ideals). Similar results are also obtained for the free unitary associative k-algebras. The proof is based on the observation that there is a natural bijection between the sets of maximal T-spaces of the free associative $k$-algebras over a nonempty set X and over a singleton set. This permits the transfer of results from the study of the lattice of T-spaces of the free associative k-algebra over a one-element set to the general case.