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 aIt 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.be a commutative associative ring with identity element 1, usually a field.Then the above rule implies that the monomials are ordered in the following way:Any element shares an equivalence class modulo[1] The idea of non-commutative Gröbner bases is to find a set of generatorsBergman's Diamond Lemma lets us verify if a set of generatorsMoreover, 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.is called reduction-unique if given two finite compositions of reductionsIn 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.This is called an overlap ambiguity, because there are two possible reductions, namelycan be reduced to a common expression using compositions of reductions.[1] 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.the leading words under some fixed admissible ordering ofThen the following are equivalent: Here the reductions are done with respect to the fixed set of generators, which is the quantum polynomial ring in 3 variables, and assumeto be degree lexicographic order, then the leading words of the defining relations areThus the ambiguity resolves and the Lemma implies thatUnder the same ordering as in the previous example, the leading words of the generators of the ideal areThe following short algorithm follows from the proof of Bergman's Diamond Lemma.It is based on adding new relations which resolve previously unresolvable ambiguities.are distinct linear combinations of reduced words.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[1] Now, the previously unresolvable overlap ambiguity resolves by construction of the new relation.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.Since the coefficient of the leading word is -1 we scale the relation and then addNow all ambiguities resolve and Bergman's Diamond Lemma implies that