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21
Multiplicative bases, Gröbner bases, and right Gröbner bases
 JOURNAL OF SYMBOLIC COMPUTATION
, 2000
"... In this paper, we study conditions on algebras with multiplicative bases so that there is a Gröbner basis theory. We introduce right Gröbner bases for a class of modules. We give an elimination theory and intersection theory for right submodules of projective modules in path algebras. Solutions to h ..."
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Cited by 22 (3 self)
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In this paper, we study conditions on algebras with multiplicative bases so that there is a Gröbner basis theory. We introduce right Gröbner bases for a class of modules. We give an elimination theory and intersection theory for right submodules of projective modules in path algebras. Solutions to homogeneous systems of linear equations with coefficients in a quotient of a path algebra are studied via right Gröbner basis theory.
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The tech ..."
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Cited by 15 (5 self)
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The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
Algorithms and Orders for Finding Noncommutative Gröbner Bases
, 1997
"... The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expe ..."
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Cited by 12 (1 self)
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The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expensive, and in noncommutative algebras is not guaranteed to terminate. The algorithm, together with the order used to determine the leading term of each polynomial, are known to affect the cost of the computation, and are the focus of this thesis. A Gröbner basis is a set of polynomials computed, using Buchberger's algorithm, from another set of polynomials. The noncommutative form of Buchberger's algorithm repeatedly constructs a new polynomial from a triple, which is a pair of polynomials whose leading terms overlap and form a nontrivial common multiple. The algorithm leaves a number of details underspecified, and can be altered to improve its behavior. A significant improvement is the devel...
A Polycyclic Quotient Algorithm
, 1996
"... Suppose G is a group given by a finite presentation. Let G (n) denote the nth term in the derived series of G. In 1981, Baumslag, Cannonito and Miller described an algorithm to determine whether the quotient G=G (n) is polycyclic and to find G=G (n) when it is polycyclic. However, in its ori ..."
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Cited by 10 (1 self)
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Suppose G is a group given by a finite presentation. Let G (n) denote the nth term in the derived series of G. In 1981, Baumslag, Cannonito and Miller described an algorithm to determine whether the quotient G=G (n) is polycyclic and to find G=G (n) when it is polycyclic. However, in its original form, the algorithm is not practical for computer implementation. Here, a practical algorithm to determine whether G=G (n) is polycyclic and to find it when it is polycyclic is described. The algorithm involves a generalization of the Grobner basis method of commutative ring theory to the integral group ring of a polycyclic group. An implementation of this algorithm in the language C has been developed and its efficiency is encouraging.
Computer assistance for discovering formulas in system engineering and operator theory
 Journal of Functional Analysis
, 1999
"... The objective of this paper is twofold. First we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator, matrix, and linear systems engineering theory. Since the methodology allows limited human intervention, it ..."
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Cited by 9 (2 self)
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The objective of this paper is twofold. First we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator, matrix, and linear systems engineering theory. Since the methodology allows limited human intervention, it is slightly less rigid than an algorithm. We call it a strategy. The second objective is to illustrate the methodology by deriving four theorems. The presentation of the methodology is carried out in three steps. The first step is introducing an abstraction of the methodology which we call an idealized strategy. This abstraction facilitates a high level discussion of the ideas involved. Idealized strategies cannot be implemented on a computer. The second and third steps introduce approximations of these abstractions which we call prestrategy and strategy, respectively. A strategy is more general than a prestrategy and, in fact, every prestrategy is a strategy. The above mentioned approximations are implemented on a computer. We stress that, since there is a computer implementation, the reader can use these techniques to attack their own algebra problems. Thus the paper might be of both practical and theoretical interest
GröbnerShirshov bases for categories
 Nankai Series in Pure, Applied Mathematics and Theoretical Physical, Operads and Universal Algebra
"... This paper devotes to GröbnerShirshov bases for small categories (all categories below are supposed to be small) presented by a graph (=quiver) and defining relations (see, ..."
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Cited by 8 (3 self)
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This paper devotes to GröbnerShirshov bases for small categories (all categories below are supposed to be small) presented by a graph (=quiver) and defining relations (see,
Homological aspects of semigroup gradings on rings and algebras
 MR 1701322, Zbl 0934.16038
, 1999
"... Abstract. This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup . Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the graded Cartan end ..."
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Cited by 5 (1 self)
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Abstract. This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup . Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert series in the associated path incidence ring. The rationality of theEuler characteristic, the Hilbertseries and the PoincaréBettiseries is studied when is torsionfree commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.
Noncommutative Gröbner Bases: A Computational and Theoretical Tool
 Mexico State University
, 1996
"... ..."
Gröbner Basis Cryptosystems
 AAECC
, 2005
"... In the first sections we extend and generalize Gröbner basis theory to submodules of free right modules over monoid rings. Over free monoids, we adapt the known theory for right ideals and prove versions of Macaulay’s basis theorem, the Buchberger criterion, and the Buchberger algorithm. Over monoid ..."
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Cited by 3 (0 self)
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In the first sections we extend and generalize Gröbner basis theory to submodules of free right modules over monoid rings. Over free monoids, we adapt the known theory for right ideals and prove versions of Macaulay’s basis theorem, the Buchberger criterion, and the Buchberger algorithm. Over monoids presented by a finitely generated convergent string rewriting system we generalize Madlener’s Gröbner basis theory based on prefix reduction from right ideals to right modules. After showing how these Gröbner basis theories relate to classical grouptheoretic problems, we use them as a basis for a new class of cryptosystems that are generalizations of the cryptosystems described in [2] and [8]. Well known cryptosystems such as RSA, ElGamal, Polly Cracker, Polly Two and a braid group cryptosystem are shown to be special cases. We also discuss issues related to the security of these Gröbner basis cryptosystems.
Looking for Gröbner Basis Theory for (Almost) Skew 2Nomial Algebras
, 2008
"... In this paper, we introduce (almost) skew 2nomial algebras, establish the existence of a skew multiplicative Kbasis for a skew 2nomial algebra, and explore the existence of a Gröbner basis theory for such algebras. ..."
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Cited by 2 (2 self)
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In this paper, we introduce (almost) skew 2nomial algebras, establish the existence of a skew multiplicative Kbasis for a skew 2nomial algebra, and explore the existence of a Gröbner basis theory for such algebras.