T. Mora (1994). An introduction to commutative and non-commutative Grobner bases. Theor. Comp. Sci.134, 131-173. 12  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and (iv) to the previous ones follows from the observation that S (A) A) The last observation is immediate, as (I) I. 2 In many applications, one wants to describe a Gr obner basis for J , preferably related to a Gr obner basis for I. Let us recall quickly what those bases are (see [9, 13] for an introduction to the subject) We will say term to mean either word or monomial , so we can treat commutative and noncommutative polynomials simultaneously. A necessary ingredient for a Gr obner bases theory is to x a term order : a total order on the terms, compatible with ....

T. Mora, An introduction to commutative and noncommutative Grobner bases, Theoret. Comput. Sci., 134 (1994) 131-173.

....this section we review the basic definitions for noncommutative Grobner bases and rewriting systems, as well as the theorem (Theorem 1.7) connecting these two concepts. For further information about rewriting systems see [11] for a more detailed account of noncommutative Grobner bases see [4] and [9], and for the connections between these topics see [7] We begin with rewriting systems. Definition 1.1 (Division ordering) Let A be a finite set and let A # be the collection of all (noncommutative) words, including the empty word, over the alphabet A. A division ordering on A # is a partial ....

Teo Mora. An introduction to commutative and noncommutative Grobner bases. Theoret. Comput. Sci., 134(1):131--173, 1994.

.... x i r OE x j 1 x j 2 Delta Delta Delta x j s or (x i 1 Delta Delta Delta x i r = x j 1 Delta Delta Delta x j s and y i 1 Delta Delta Delta y i r is before y j 1 Delta Delta Delta y j r lexicographically) For an introduction to non commutative term orders and Grobner bases see [Mo]. We shall prove that the non commutative ideal J has a quadratic Grobner basis. Theorem 3.2. Let be a quasi poset monoid and OE a term order such that in OE (I ) is quasi poset. If l is the induced non commutative term order, then in l (J ) h f y i y j j j i g [ f y i y j j i j and x i ....

T. Mora, An introduction to commutative and noncommutative Grobner bases, Theoretical Computer Science 134 (1994), 131--173.

....are mapped onto is called the non commutative fiber of . The monomials in the non commutative fiber of are in one to one correspondence with the facets ( maximal faces) in Delta( In Section 3 we prove that, if supports a poset, then the two sided ideal J has a quadratic Grobner basis (cf. [Mo]) From this non commutative Grobner basis we construct a non pure shelling for all complexes Delta( By [BW] the existence of such a shelling implies that Delta( is homotopy equivalent to a wedge of spheres of various dimensions. We determine these dimensions in Corollary 3.7. This yields a ....

.... x i r OE x j 1 x j 2 Delta Delta Delta x j s or (x i 1 Delta Delta Delta x i r = x j 1 Delta Delta Delta x j s and y i 1 Delta Delta Delta y i r is before y j 1 Delta Delta Delta y j r lexicographically) For an introduction to non commutative term orders and Grobner bases see [Mo]. We shall prove that the non commutative ideal J has a quadratic Grobner basis. Theorem 3.2. Let be a quasi poset monoid and OE a term order such that in OE (I ) is quasi poset. If l is the induced non commutative term order, then inl (J ) h f y i y j j j i g [ f y i y j j i j and x i ....

T. Mora, An introduction to commutative and noncommutative Grobner bases, Theoretical Computer Science 134 (1994), 131--173.

....Eciency can be gained in three di erent ways: 1. Avoiding the computation and reduction of overlap relations that reduce to zero. Buchberger [5] stated several criteria that allow this prediction in commutative algebras, and similar results have been shown for the free noncommutative case [14, 15, 11]. In the noncommutative case, there is one criterion on the common multiple formed by the leading terms of the polynomials that make up a overlap relation. Simpli ed slightly, the criterion is that if the common multiple is divisible by the leading term of another polynomial in the Gr obner basis, ....

Mora T. An introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science 134, 1 (1994) 131-173.

....# 0 , then the set of matrices M# (G) is contained in the closure of the interior of M# (G) 8 Main Theorems on Convexity and Positivity 8.1 Main result on convexity: Theorem 8. 2 The next theorem gives a test which can in fact be implemented with a noncommutative Grobner basis algorithm (see [Mor86, Mor94] and [Fro97] The linear dependence check is purely algebraic and can be performed automatically by computer (software willing) We have not considered seriously the practicality of the Openness Property. However, in all examples we have done it is obvious that the sets G obtained satisfy it. ....

Teo Mora. An introduction to commutative and noncommutative Grobner bases. Theoretical Computer Science, 134(1):131--173, 7 November 1994.

....important results, contains many more references of this note and an advanced discussion of the complexity of system solving. Introduction Recalls on Grobner bases Since Grobner bases will be mainly used in this paper, let us introduce the notation and definition we need, refering the results to [Mo] Let T be the commutative semigroup generated by fX 1 ; Xng and let us assume that it is endowed with a well ordering , which is a semigroup ordering, i.e. it is compatible with the product: for each t; t 1 ; t 2 2 T; t 1 t 2 implies t t 1 t t 2 We consider also the polynomial ring ....

T. Mora, An introduction to commutative and non-commutative Grobner bases, Theor. Comp. Sci., to appear

....members of IC[C 0 but which are not members of IC 0 . Categorize is not implementable on a computer, because subsumed in the Categorize command is the ability to eliminate unknowns (whenever algebraically possible) from equations in the original set of polynomial equations. In fact, the paper [TMora] shows that there does not exist a computer algorithm which can determine whether or not P 1 is the empty set. For an example of how Categorize can be useful, suppose that C is a collection of polynomial equations in knowns a j and unknowns x k . If it could be shown algebraically that the Riccati ....

....simplification of complicated expressions. As we shall see, the present paper concerns a different type of application, that of eliminating unknowns from collections of equations, which is the main function of the NCProcess commands. In this paper, Grobner Bases are computed using an algorithm in [TMora]. See also x10 for further discussion. We use the abbreviations GB and GBA to refer to Grobner Basis and Grobner Basis Algorithm respectively. Since we will not always let the GBA run until it finds a Grobner Basis, we will often be dealing with sets which are not Grobner Bases, but rather ....

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T. Mora, "An introduction to commutative and noncommutative Grobner Bases", Theoretical Computer Science, Nov 7,1994, vol. 134 N1:131-173.

....to G such that t i j xHT(g i )y for some x; y 2 Sigma , i.e. HT(g i ) divides t i . Note that as soon as all such g i are added to G, we have HT(ideal(G) S k i=1 H t i and all further computed s polynomials must reduce to zero (we take the notion of s polynomials as defined by Mora in [Mo94]) Since the procedure is correct, G then is also a Grobner basis of ideal(F ) As we have seen before, procedure Gr obner Bases in Free Monoid Rings on input P T only produces new polynomials of the form 0, u Gamma v or Gammau v. Hence on termination the output has the desired form. q.e.d. ....

....relations called prefix respectively suffix reduction can be defined and finite prefix respectively suffix Grobner bases of the right respectively left ideals exist. These bases can be in fact computed by interreducing the generating set with respect to prefix or suffix reduction (compare e.g. [Mo94] for an algorithm to compute prefix Grobner bases for finitely generated right ideals) As stated in the introduction, a first explicit connection between finitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution to the word problem in ....

T. Mora. An Introduction to Commutative and Non-Commutative Grobner Bases. Theoretical Computer Science 134(1994). pp 131-173. 29

....Many packages exist for computing Gr obner bases for commutative algebras including Groebner by Windsteiger and Buchberger [3] Non commutative algebras present more diculties. Computing a Gr obner basis for a non commutative algebra is a process that is not guaranteed to terminate, see Mora [25]) However, packages do exist to compute Gr obner bases or partial Gr obner bases for non commutative algebras, including Opal by Keller [24] 17 4.2 Extracting the Simple Components Extracting the simple components of an algebra is the next major step in decomposition. In this section, all ....

Teo Mora. An introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science, (134):131-173, 1994.

....divisors. The theory of Grobner bases has been extended to certain classes of algebras with these properties (e.g. Mora 1986, Apel 1988, Kandri Rody Weispfenning 1990, Apel 1992, Kredel 1993, Madlener Reinert 1993) including the algebras of interest here. A recent survey has been given by Mora (1994). This paper discusses one such extension, and presents algorithms optimized for these algebras. Since the algebras are non commutative, one sided and two sided ideals do not coincide. However, our work is concerned solely with one sided ideals, since these suffice for our applications (see ....

Mora, F. (1994). An introduction to commutative and non-commutative Grobner bases. J. Theoretical Comp. Sci. 135, 131--174.

....monomials in khy 1 ; yn i which are mapped onto z is called the non commutative fiber of . The monomials in the non commutative fiber of are in one to one correspondence with the facets ( maximal faces) in Delta( ffl In Theorem 3. 5 we relate non commutative Grobner basis (see [Mo]) to a nonpure shelling for all complexes Delta( By [BW] the existence of such a shelling implies that Delta( is homotopy equivalent to a wedge of spheres of various dimensions. Theorem 3.5 gives a shelling of the monoid, that is a uniform rule for simultaneously shelling all finite ....

T. Mora, An introduction to commutative and noncommutative Grobner bases, Theoretical Computer Science 134 (1994), 131--173.

....to transform a finite generating set of a polynomial ideal into a finite Grobner basis of the same ideal. Since the theory of Grobner bases turned out to be of outstanding importance for polynomial rings, extensions of Buchberger s ideas to other algebras followed, for example to free algebras ([Mo85, Mo94]) Weyl algebras ( La85] enveloping fields of Lie algebras ( ApLa88] solvable rings ( KaWe90, Kr93] skew polynomial rings ( We92] free group rings ( Ro93] and monoid and group rings ( MaRe93b] Especially group rings are subject of extensive studies in mathematics. In 1981 Baumslag, ....

T. Mora. An Introduction to Commutative and Non-Commutative Grobner Bases. Theoretical Computer Science 134(1994). pp 131-173.

....special classes of non commutative algebras, e.g. Weyl algebras and later on Lie algebras [2] This works because the structures studied allow representations by commutative polynomials for their elements. Unfortunately this approach is not possible for general non commutative algebras. Mora [16, 17] generalized Grobner bases for noncommutative polynomial rings, i.e. free monoid rings, where multiplication of terms is just concatenation of words. An implementation of his procedure has for example been done in the system Opal by Keller [7] The next step of generalizing Grobner bases was to ....

T. Mora. An introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science, 134:131--173, 1994.

.... investigation due to the great theoretical and practical importance To be published in Mathematics and Computers in Simulation of these bases in computational commutative algebra and algebraic geometry [2, 3, 4] Grobner bases are also becoming of greater importance in non commutative [5, 6, 7] and differential algebra [8, 9] Since its invention about thirty years ago, feasibility of the Buchberger algorithm has been notably increased. First of all, it was resulted from discovering criteria for avoiding unnecessary reductions [10, 11, 12] which allow a partial extension to ....

Mora, T. (1994). An Introduction to Commutative and Non-Commutative Grobner Bases. Theor. Comp. Sci. 134, 131-173.

....basis computations, progress has been made toward improving the e#ciency through algorithms that eliminate unnecessary work. However, for Grobner bases in noncommutative algebras, all that is known (or speculated) is that the commutative techniques should still apply (see the survey by Mora [46]) The algorithms considered in this research are a mix of new algorithms and adaptations of algorithms used in the commutative case. We successfully identify a configuration of alternative algorithms that computes noncommutative Grobner bases more e#ciently. The second factor is the choice of ....

....Y Z, which is not divisible by either element of the generating set and so is not further reducible. Benjamin J. Keller Chapter 2 14 Therefore, given an arbitrary generating set P for an ideal I, di#erent reductions may yield di#erent results. In fact, ideal membership is in general undecidable [35, 46]. However, for decidable instances (including all instances in commutative polynomial rings) if a finite generating set P is given, then it is possible to find another generating set GP which generates the same ideal and for which reduction always yields a unique value. Such a generating set is ....

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T. Mora. An introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science, 134:131--173, 1994.

....fl Gamma1 (I) This is especially true in the construction of free resolutions as in [An] Non commutative presentations have been exploited in [AR] and [PRS] to study homology of coordinate rings of Grassmannians and toric varieties. These applications all make use of Grobner bases for J (see [Mo] for non commutative Grobner bases. In this note we give an explicit description (Theorem 2.1) of the minimal Grobner bases for J with respect to monomial orders on khXi that are lexicographic extensions of monomial orders on k[x] Non commutative Grobner bases are usually infinite; for example, ....

T. Mora, An introduction to commutative and non-commutative Grobner bases, Theoretical Computer Science 134 (1994), 131--173.

....strictly increasing sequence of left ideals. Therefore, not all left ideals have a finite 18 Fr ed eric Chyzak and Bruno Salvy basis y . However, it is possible to extend slightly the ideas in (Kandri Rody and Weispfenning, 1990) so as to accomodate Ore algebras. We refer the reader to (Mora, 1994) for a survey on Grobner bases. As in the commutative case, Grobner bases are defined with respect to admissible term orders, which makes it possible to generalize the leading term used in the Euclidean division. This is obtained by considering the set T of terms in the algebra. Each algebra for ....

Mora, T. (1994). An introduction to commutative and noncommutative Grobner bases. Theoretical Computer Science, 134:131--173.

....not produce simplification in the same sense as the graded lexicographic orders used in the earlier models. In particular, it does not reduce the number of factors in terms. It should also be noted that there are finitely generated ideals which have an infinite Grobner Basis for every ordering [TMora]. 4.2 Infinite Bases and Transformations Model E in section 3 provides another way in which infinite families can occur. Notice that without the two relations involving h(xy) and h(yx) John J Wavrik this is the same as Model B, which has a finite basis. The introduction of two new variables ....

. Mora, "An introduction to commutative and noncommutative Grobner Bases", Theoretical Computer Science, vol.134 (1994) pp.131--173.

....greatest common divisor of the nonzero polynomials in S and can be found by the Euclidean algorithm. In fact, Buchberger s algorithm for computing Grobner bases can be thought of as a multivariate version of the Euclidean algorithm. Two good introductory references on Grobner bases are [BW91] and [Mor94]. The generalization of Grobner bases to noncommutative rings is an area of active research. Let khXi denote the free associative algebra over the field k with X as the set of indeterminates. The natural question is: can Buchberger s algorithm be generalized to khXi The answer to this question is ....

T. Mora. An introduction to commutative and non-commutative Grobner bases. Theoretical Computer Science, 134(1):131--174, 1994.

....of Dickson s lemma. The usual polynomial ring can be viewed as a monoid ring where the monoid is a finitely generated free commutative monoid. Mora studied the class where the free commutative monoid is substituted by a free monoid the class of finitely generated free monoid rings (compare e.g. [Mo85, Mo94]) The ring operations are mainly performed in the coefficient domain while the terms are treated like words, i.e. the variables no longer commute with each other. The definitions of (one and two sided) ideals, reduction and Grobner bases are carried over from the commutative case to establish a ....

T. Mora. An Introduction to Commutative and Non-Commutative Grobner Bases. Theoretical Computer Science 134(1994). pp 131-173.

....to transform a finite generating set of a polynomial ideal into a finite Grobner basis of the same ideal. Since the theory of Grobner bases turned out to be of outstanding importance for polynomial rings, extensions of Buchberger s ideas to other algebras followed, for example to free algebras ([Mo85, Mo94]) Weyl algebras ( La85] enveloping fields of Lie algebras ( ApLa88] solvable rings ( KaWe90, Kr93] skew polynomial rings ( We92] free group rings ( Ro93] and monoid and group rings ( MaRe93b] In [MaRe93a] we have combined the ideas of string rewriting and polynomial rewriting in the ....

T. Mora. An Introduction to Commutative and Non-Commutative Grobner Bases. Theoretical Computer Science 134(1994). pp 131-173.

No context found.

T. Mora (1994). An introduction to commutative and non-commutative Grobner bases. Theor. Comp. Sci.134, 131-173. 12

.... , we will denote T (F ) f T (f) 2 T; f 2 F; f 6= 0g: Setting U : ker( and N(U) T n T (U) we have = U V; V = P =U = Span K (N(U) Thanks to these isomorphisms, we can consider T : N(U) as a K basis of V so that the (linear algebra) Gr obner technology discussed in [23] and [20] can be projected from P to V . In particular, T can be ordered (only as a set ) by and each element f 2 V c i i ; c i 2 K n f0g; i 2 T ; 1 2 and we can denote T (f) 1 its maximal term. Lemma 1.5 8f 2 V; 8j; X j T (f) 2 T = T (X j (f) X j T (f) ....

Mora T. An Introduction to Commutative and Noncommutative Grobner Bases. Th. Comp. Sci. 1994, 134, 131-173.

....Can(f; I; of f modulo I) Can(f; I; Can(f; I) is called the canonical (normal) form of f modulo I (and the dependence on is omitted if no confusion arise) The following definitions and results (1.3, 1.4, and 1. 5) belong to the noncommutative Grobner bases theory on free algebras cf. [M2]. Theorem 1.3. Some characterizations of Grobner bases. Let F ae I n f0g. Then, the following properties are equivalent: i. T (F ) T (I) ii. N (F ) is a K basis of SpanK (N (I) iii. f(s) j s 2 N (F )g is a K basis of KhXi=I, where : KhXi KhXi=I is the canonical projection. A ....

....basis leads to an algorithm that allows us to multiply words in canonical form in 5 steps at most (cf. 1.9.ii) 5. Further generalization 5.1. FGLM algorithm for semigroup rings In fact, it can be possible to build, when a semigroup S is given, Grobner basis tools for K[S] cf. section 7 of [M2]) Therefore, we are going to discuss herein key ideas in order to design an algorithm like FGLM for two sided ideals of K[S] There are some essential points in algorithms like FGLM, namely: ffl Explore a certain finite subset T of S where all the heads of rGb(I) have to be contained. ....

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Mora T. (1994). "An Introduction to Commutative and Noncommutative Grobner Bases." Theoretical Computer Science 134, pp. 131-173.

.... , we will denote T (F ) f T (f) 2 T; f 2 Fg: Setting U : ker( and N(U) T n T (U) we have P = U V; V = P =U = Span K (N(U) Thanks to these congruences, we can consider T : N(U) as a K basis of V so that the (linear algebra) Gr obner technology discussed in [13] and [11] can be projected from P to V . In particular, T can be ordered (as a set only ) by and each element f 2 V can be uniquely expressed as f = X c i i ; c i 2 K n f0g; i 2 T; 1 2 and we can denote T (f) 1 its maximal term. Lemma 1.2 8f 2 V; 8j; X j T (f) 2 ....

Mora T. An Introduction to Commutative and Noncommutative Grobner Bases. Th. Comp. Ssci. 134 (1994), 131-173.

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Teo Mora. An introduction to commutative and noncommutative Grobner bases. Theoretical Computer Science, 134(1):131-173, 7 November 1994.

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