scholarly journals Monomial Orderings, Rewriting Systems, and Gröbner Bases for the Commutator Ideal of a Free Algebra

1999 ◽  
Vol 27 (2) ◽  
pp. 133-141 ◽  
Author(s):  
S.M. Hermiller ◽  
X.H. Kramer ◽  
R.C. Laubenbacher
Keyword(s):  
Free Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Commutator Ideal ◽  
Rewriting Systems

NONCOMMUTATIVE GRÖBNER BASES FOR THE COMMUTATOR IDEAL

2006 ◽  
Vol 16 (01) ◽  
pp. 187-202 ◽  
Author(s):  
SUSAN HERMILLER ◽  
JON McCAMMOND
Keyword(s):  
Associative Algebra ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Free Associative Algebra ◽  
Arbitrary Field ◽  
Grobner Basis ◽  
Commutator Ideal ◽  
Universal Gröbner Basis

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I, and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis.

The General PBW Property

2007 ◽  
Vol 14 (04) ◽  
pp. 541-554
Author(s):  
Huishi Li
Keyword(s):  
Free Algebra ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Path Algebra ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Graded Ring ◽  
Homological Characterization

For ungraded quotients of an arbitrary ℤ-graded ring, we define the general PBW property, that covers the classical PBW property and the N-type PBW property studied via the N-Koszulity by several authors (see [2–4]). In view of the noncommutative Gröbner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [5] concerning Gröbner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.

scholarly journals Gröbner bases for operads

2010 ◽  
Vol 153 (2) ◽  
pp. 363-396 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Anton Khoroshkin
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases
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