| ### 7.9.3 Groebner bases for two-sided ideals in free associative algebras We say that a monomial 331#331 divides (two-sided or bilaterally) a monomial 346#346, if there exist monomials 398#398, such that 399#399, in other words 331#331 is a subword of 346#346. Let 400#400 be the free algebra and $<$ be a fixed monomial ordering on $T$. For a subset 401#401, define the leading ideal of 189#189 to be the two-sided ideal 402#402 403#403 404#404. A subset 250#250 is a (two-sided) Groebner basis for the ideal 251#251 with respect to 226#226, if 405#405. That is 406#406 there exists 254#254, such that 407#407 divides 408#408. 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 409#409. In other words, by trimming elements of degree exceeding 171#171 from the complete Groebner basis 189#189, one obtains precisely 409#409. In general, given a set 409#409, 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 410#410 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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