| ### 7.9.3 Groebner bases for two-sided ideals in free associative algebras We say that a monomial 333#333 divides (two-sided or bilaterally) a monomial 348#348, if there exist monomials 400#400, such that 401#401, in other words 333#333 is a subword of 348#348. Let 402#402 be the free algebra and $<$ be a fixed monomial ordering on $T$. For a subset 403#403, define the leading ideal of 190#190 to be the two-sided ideal 404#404 405#405 406#406. A subset 252#252 is a (two-sided) Groebner basis for the ideal 253#253 with respect to 228#228, if 407#407. That is 408#408 there exists 256#256, such that 409#409 divides 410#410. The notion of Groebner-Shirshov basis applies to more general algebraic structures, but means the same as Groebner basis for associative algebras. Suppose, that the weights of the ring variables are strictly positive. We can interpret these weights as defining a non-standard grading on the ring. If the set of input polynomials is weighted homogeneous with respect to the given weights of the ring variables, then computing up to a weighted degree (and thus, also length) bound 171#171 results in the truncated Groebner basis 411#411. In other words, by trimming elements of degree exceeding 171#171 from the complete Groebner basis 190#190, one obtains precisely 411#411. In general, given a set 411#411, which is the result of Groebner basis computation up to weighted degree bound 171#171, then it is the complete finite Groebner basis, if and only if 412#412 holds. Note: If the set of input polynomials is not weighted homogeneous with respect to the weights of the ring variables, and a Groebner is not finite, then actually not much can be said precisely on the properties of the given ideal. By increasing the length bound bigger generating sets will be computed, but in contrast to the weighted homogeneous case some polynomials in of small length first enter the basis after computing up to a much higher length bound.

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