Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

TL;DR

This paper extends the letterplace correspondence to nongraded noncommutative ideals, enabling the modeling of such algebras via graded commutative algebras and introducing algorithms for computing inhomogeneous noncommutative Gröbner bases.

Contribution

It generalizes the letterplace correspondence to nongraded ideals and develops new algorithms using homogeneous commutative polynomials for noncommutative algebra computations.

Findings

Extended letterplace correspondence to nongraded ideals

Developed saturation notions for graded ideals and their analogues

Implemented algorithms in Maple and Singular for practical validation

Abstract

Let $K\ < x_i\ >$ be the free associative algebra generated by a finite or countable number of variables $x_i$. The notion of "letterplace correspondence" introduced in [1,2] for the graded (two-sided) ideals of $K\ < x_i\ >$ is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid $\mathbb N$ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of $K\ < x_i,t\ >$, where $t$ is an extra variable and for their letterplace analogues in the commutative polynomial algebra $K[x_{ij},t_j]$, where $j$ ranges in $\mathbb N$. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gr\"obner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular. References [1] La Scala, R.; Levandovskyy, V., Letterplace ideals and non-commutative Gr\"obner bases. J. Symbolic Comput., 44 (2009), 1374--1393. [2] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gr\"obner bases and the letterplace embedding of the free associative algebra. J. Symbolic Comput., 48 (2013), 110--131