The ``Freedom Theorem" of Left-Commutative Algebras

羽 李

Abstract

Let X be a finite set and LC(X)

b e t h e f r e e l e f t - c o m m u t a t i v e a l g e b r a i n d u c e d b y X . L e t I d (f)

$be the free left-commutative algebra induced by X. Let Id(f)$ be the ideal of LC(X) induced by f where

f \in L C (X)

$f\in LC(X)$ . For any

h

$h$ , the problem is whether there is an algorithm to decide

h \in I d (f)

$h\in Id(f)$ or

h \notin I d (f)

$h\notin Id(f)$ . This problem is studied by using the approach of Grobner-Shirshov bases theory. A well ordering on a linear basis of free left commutative algebra is defined. It is proved that the ordering is compatible with the product and that the element of the ideal of free left-commutative algebra induced by one polynomial is rewritten. The word problem for left-commutative algebras with a single defining relation is solved and the ``freedom theorem for left-commutative algebras is obtained.

Keywords:

word problem

References

[1] L.A. Bokut and G.P. Kukin. Algorithmic and combinatorial algebra., Mathematics and its Applications 255. Kluwer Academic Publishers Group, Dordrecht, 1994.
[2] A. Dzhumadil'daev and C. L\"{o}fwall. Trees, free right-symmetric algebras, free Novikov algebras and identities. Homology, Homotopy and Applications. 4(2)(2002), 165–190.
[3] D. Kozybaev, L. Makar-Limanov and U. Umirbaev. The freiheitssatz and automorphisms of free right-symmetric algebras. Asian-European Journal of Mathematics.
1(2)(2008), 243-254.
[4] J.-L. Loday. Cup product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77, Univ. Louis Pasteur, Strasbourg, 1995, 189-196.
[5] W. Magnus. \"{U}ber discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssaz). J. Reine Angew. Math. 163(1930), 141-165.
[6] L. Makar-Limanov. Algebraically closed skew fields. J. Algebra 93(1985), 117-135.
[7] N.S. Romanovskii. A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees.(Russian) Mat. Sb. (N.S.) 89(131)(1972), 93-99.
[8] A.I. Shirshov. Some algorithmic problems for Lie algebras. Sibirsk. Mat. Z. 3(1962), 292-296.

The ``Freedom Theorem" of Left-Commutative Algebras

Abstract

References

Catalog

Export File

Citation

Format

Content