2012/06/26 by Roberto La Scala, La Scala, Roberto
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1206.6027
openalex publication_date 2012/06/26 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Let K < xi > be the free associative algebra generated by a finite or\ncountable number of variables xi. The notion of "letterplace correspondence"\nintroduced in [1,2] for the graded (two-sided) ideals of K < xi > is\nextended in this paper also to the nongraded case. This amounts to the\npossibility of modelizing nongraded noncommutative presented algebras by means\nof a class of graded commutative algebras that are invariant under the action\nof the monoid mathbb N of natural numbers. For such purpose we develop the\nnotion of saturation for the graded ideals of K < xi,t >, where t is an\nextra variable and for their letterplace analogues in the commutative\npolynomial algebra K[xij,tj], where j ranges in mathbb N. In\nparticular, one obtains an alternative algorithm for computing inhomogeneous\nnoncommutative Gr "obner bases using just homogeneous commutative polynomials.\nThe feasibility of the proposed methods is shown by an experimental\nimplementation developed in the computer algebra system Maple and by using\nstandard routines for the Buchberger algorithm contained in Singular.\n References\n [1] La Scala, R.; Levandovskyy, V., Letterplace ideals and non-commutative\nGr "obner bases. J. Symbolic Comput., 44 (2009), 1374--1393.\n [2] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gr "obner bases\nand the letterplace embedding of the free associative algebra. J. Symbolic\nComput., 48 (2013), 110--131\n