FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2001, VOLUME 7, NUMBER 2, PAGES 495-513
D. I. Piontkovsky
Abstract
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In the paper we consider a class of associative algebras
which are denoted by \emph{algebras with $R$ -processing}.
This class includes free algebras, finitely-defined
monomial algebras, and semigroup algebras for some monoids.
A sufficient condition for $A$ to be an algebra
with $R$ -processing is formulated in terms of a special graph,
which includes a part of information about overlaps
between monomials forming the reduced Gr\"obner
basis for a syzygy ideal of $A$
(for monoids,
this graph includes the information about overlaps between
right and left parts of suitable string-rewriting system).
Every finitely generated right ideal in an algebra
with $R$ -processing has a finite Gr\"obner basis,
and the right syzygy module of the ideal is finitely generated,
i. e. every such algebra is coherent.
In such algebras,
there exist algorithms for computing a Gr\"obner basis
for a right ideal, for the membership test for
a right ideal, for zero-divisor test, and for
solving systems of linear equations.
In particular, in a monoid with $R$ -processing there exist
algorithms for word equivalence test and for
left-divisor test as well.
All articles are published in Russian.
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Last modified: October 31, 2001.