Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms
arXiv:1206.6027
Abstract
Let be the free associative algebra generated by a finite or countable number of variables . The notion of "letterplace correspondence" introduced in [1,2] for the graded (two-sided) ideals of 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 of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of , where is an extra variable and for their letterplace analogues in the commutative polynomial algebra , where ranges in . In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner 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 Gröbner bases. J. Symbolic Comput., 44 (2009), 1374--1393. [2] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra. J. Symbolic Comput., 48 (2013), 110--131
arXiv comment22 pages, to appear in International Journal of Algebra and Computation