Bergman's diamond lemma

In mathematics, specifically the field of abstract algebra, Bergman's Diamond Lemma (after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a -basis. It is an extension of Gröbner bases to non-commutative rings. The proof of the lemma gives rise to an algorithm for obtaining a non-commutative Gröbner basis of the algebra from its defining relations. However, in contrast to Buchberger's algorithm, in the non-commutative case, this algorithm may not terminate.[1]

Preliminaries

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Let   be a commutative associative ring with identity element 1, usually a field. Take an arbitrary set   of variables. In the finite case one usually has  . Then   is the free semigroup with identity 1 on  . Finally,   is the free associative  -algebra over  .[2][3] Elements of   will be called words, since elements of   can be seen as letters.

Monomial Ordering

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The reductions below require a choice of ordering   on the words i.e. monomials of  . This has to be a total order and satisfy the following:

  1. For all words   and  , we have that if   then  .[2]
  2. For each word  , the collection   is finite.[1]

We call such an order admissible.[4] An important example is the degree lexicographic order, where   if   has smaller degree than  ; or in the case where they have the same degree, we say   if   comes earlier in the lexicographic order than  . For example the degree lexicographic order on monomials of   is given by first assuming  . Then the above rule implies that the monomials are ordered in the following way:

 

Every element   has a leading word which is the largest word under the ordering   which appears in   with non-zero coefficient.[1] In   if  , then the leading word of   under degree lexicographic order is  .

Reduction

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Assume we have a set   which generates a 2-sided ideal   of  . Then we may scale each   such that its leading word   has coefficient 1. Thus we can write  , where   is a linear combination of words   such that  .[1] A word   is called reduced with respect to the relations   if it does not contain any of the leading words  . Otherwise,   for some   and some  . Then there is a reduction  , which is an endomorphism of   that fixes all elements of   apart from   and sends this to  .[2] By the choice of ordering there are only finitely many words less than any given word, hence a finite composition of reductions will send any   to a linear combination of reduced words.

Any element shares an equivalence class modulo   with its reduced form. Thus the canonical images of the reduced words in   form a  -spanning set.[1] The idea of non-commutative Gröbner bases is to find a set of generators   of the ideal   such that the images of the corresponding reduced words in   are a  -basis. Bergman's Diamond Lemma lets us verify if a set of generators   has this property. Moreover, in the case where it does not have this property, the proof of Bergman's Diamond Lemma leads to an algorithm for extending the set of generators to one that does.

An element   is called reduction-unique if given two finite compositions of reductions   and   such that the images   and   are linear combinations of reduced words, then  . In other words, if we apply reductions to transform an element into a linear combination of reduced words in two different ways, we obtain the same result.[5]

 
The series of reductions lead to a common expression. The diamond shape gives rise to the name.

Ambiguities

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When performing reductions there might not always be an obvious choice for which reduction to do. This is called an ambiguity and there are two types which may arise. Firstly, suppose we have a word   for some non-empty words   and assume that   and   are leading words for some  . This is called an overlap ambiguity, because there are two possible reductions, namely   and  . This ambiguity is resolvable if   and   can be reduced to a common expression using compositions of reductions.

Secondly, one leading word may be contained in another i.e.   for some words   and some indices  . Then we have an inclusion ambiguity. Again, this ambiguity is resolvable if  , for some compositions of reductions   and  .[1]

Statement of the Lemma

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The statement of the lemma is simple but involves the terminology defined above. This lemma is applicable as long as the underlying ring is associative.[6]

Let   generate an ideal   of  , where   with   the leading words under some fixed admissible ordering of  . Then the following are equivalent:

  1. All overlap and inclusion ambiguities among the   are resolvable.
  2. All elements of   are reduction-unique.
  3. The images of the reduced words in   form a  -basis.

Here the reductions are done with respect to the fixed set of generators   of  . When any of the above hold we say that   is a Gröbner basis for  .[1] Given a set of generators, one usually checks the first or second condition to confirm that the set is a  -basis.

Examples

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Resolving ambiguities

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Take  , which is the quantum polynomial ring in 3 variables, and assume  . Take   to be degree lexicographic order, then the leading words of the defining relations are  ,   and  . There is exactly one overlap ambiguity which is   and no inclusion ambiguities. One may resolve via   or via   first. The first option gives us the following chain of reductions,

 

whereas the second possibility gives,

 

Since   are commutative the above are equal. Thus the ambiguity resolves and the Lemma implies that   is a Gröbner basis of  .

Non-resolving ambiguities

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Let  . Under the same ordering as in the previous example, the leading words of the generators of the ideal are  ,   and  . There are two overlap ambiguities, namely   and  . Let us consider  . If we resolve   first we get,

 

which contains no leading words and is therefore reduced. Resolving   first we obtain,

 

Since both of the above are reduced but not equal we see that the ambiguity does not resolve. Hence   is not a Gröbner basis for the ideal it generates.

Algorithm

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The following short algorithm follows from the proof of Bergman's Diamond Lemma. It is based on adding new relations which resolve previously unresolvable ambiguities. Suppose that   is an overlap ambiguity which does not resolve. Then, for some compositions of reductions   and  , we have that   and   are distinct linear combinations of reduced words. Therefore, we obtain a new non-zero relation  . The leading word of this relation is necessarily different from the leading words of existing relations. Now scale this relation by a non-zero constant such that its leading word has coefficient 1 and add it to the generating set of  . The process is analogous for inclusion ambiguities.[1]

Now, the previously unresolvable overlap ambiguity resolves by construction of the new relation. However, new ambiguities may arise. This process may terminate after a finite number of iterations producing a Gröbner basis for the ideal or never terminate. The infinite set of relations produced in the case where the algorithm never terminates is still a Gröbner basis, but it may not be useful unless a pattern in the new relations can be found.[7]

Example

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Let us continue with the example from above where  . We found that the overlap ambiguity   does not resolve. This gives us   and  . The new relation is therefore   whose leading word is   with coefficient 1. Hence we do not need to scale it and can add it to our set of relations which is now  . The previous ambiguity now resolves to either   or  . Adding the new relation did not add any ambiguities so we are left with the overlap ambiguity   we identified above. Let us try and resolve it with the relations we currently have. Again, resolving   first we obtain,

 

On the other hand resolving   twice first and then   we find,

 

Thus we have   and   and the new relation is   with leading word  . Since the coefficient of the leading word is -1 we scale the relation and then add   to the set of defining relations. Now all ambiguities resolve and Bergman's Diamond Lemma implies that

  is a Gröbner basis for the ideal it defines.

Further generalisations

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The importance of the diamond lemma can be seen by how many other mathematical structures it has been adapted for:

The lemma has been used to prove the Poincaré–Birkhoff–Witt theorem.[2]

References

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  1. ^ a b c d e f g h Rogalski, D. (2014-03-12). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].
  2. ^ a b c d Bergman, George (1978-02-01). "The diamond lemma for ring theory". Advances in Mathematics. 29 (2): 178–218. doi:10.1016/0001-8708(78)90010-5. ISSN 0001-8708.
  3. ^ Dotsenko, Vladimir; Tamaroff, Pedro (2020-10-28). "Tangent complexes and the Diamond Lemma". arXiv:2010.14792 [math.RA].
  4. ^ a b Lopatkin, Viktor (2021-10-12). "Garside Theory: a Composition--Diamond Lemma Point of View". arXiv:2109.07595 [math.RA].
  5. ^ Reyes, A., Suárez, H. (2016-12-01) "Bases for Quantum Algebras and skew Poincare-Birkhoff-Witt Extensions". MOMENTO No 54. ISSN 0121-4470
  6. ^ a b Hellström, L (2002-10-22) "The Diamond Lemma for Power Series Algebras". Print & Media, Umeå universitet, Umeå. ISBN 91-7305-327-9
  7. ^ Li, Huishi (2009-06-23). "Algebras Defined by Monic Gr\"obner Bases over Rings". arXiv:0906.4396 [math.RA].
  8. ^ Elias, Ben (2019-07-24). "A diamond lemma for Hecke-type algebras". arXiv:1907.10571 [math.RT].
  9. ^ Bokut, L. A.; Chen, Yuqun; Li, Yu (2011-01-07). "Gr\"obner-Shirshov bases for categories". arXiv:1101.1563 [math.RA].
  10. ^ Dotsenko, Vladimir; Khoroshkin, Anton (2010-06-01). "Gröbner bases for operads". Duke Mathematical Journal. 153 (2): 363–396. arXiv:0812.4069. doi:10.1215/00127094-2010-026. ISSN 0012-7094. S2CID 12243016.