### 7.5.19 ncpreim_lib

Library:
  ncpreim.lib
Purpose:
  Non-commutative elimination and preimage computations
Author:
  Daniel Andres, daniel.andres@math.rwth-aachen.de

  Support: DFG Graduiertenkolleg 1632 `Experimentelle und konstruktive Algebra'

Overview:
  In G-algebras, elimination of variables is more involved than in the commutative case.
  One, not every subset of variables generates an algebra, which is again a G-algebra.
  Two, even if the subset of variables in question generates an admissible subalgebra, there might be no admissible elimination ordering, i.e. an elimination ordering which also satisfies the ordering condition for G-algebras.

  The difference between the procedure `eliminateNC` provided in this library and the procedure `eliminate (plural)` from the kernel is that eliminateNC will always find an admissible elimination if such one exists. Moreover, the use of `slimgb` for performing Groebner basis computations is possible.

  As an application of the theory of elimination, the procedure `preimageNC` is provided, which computes the preimage of an ideal under a homomorphism f: A -> B between G-algebras A and B. In contrast to the kernel procedure `preimage (plural)`, the assumption that A is commutative is not required.

References:
  (BGL) J.L. Bueso, J. Gomez-Torrecillas, F.J. Lobillo: `Re-filtering and exactness of the Gelfand-Kirillov dimension', Bull. Sci. math. 125, 8, 689-715, 2001.
  (GML) J.I. Garcia Garcia, J. Garcia Miranda, F.J. Lobillo: `Elimination orderings and localization in PBW algebras', Linear Algebra and its Applications 430(8-9), 2133-2148, 2009.
  (Lev) V. Levandovskyy: `Intersection of ideals with non-commutative subalgebras', ISSAC'06, 212-219, ACM, 2006.

Procedures:

> 7.5.19.0. eliminateNC |  | elimination in G-algebras
> 7.5.19.0. preimageNC |  | preimage of ideals under homomorphisms of G-algebras
> 7.5.19.0. admissibleSub |  | checks whether subalgebra is admissible
> 7.5.19.0. isUpperTriangular |  | checks whether matrix is (strictly) upper triangular
> 7.5.19.0. appendWeight2Ord |  | appends weight to ordering
> 7.5.19.0. elimWeight |  | computes elimination weight
> 7.5.19.0. extendedTensor |  | tensor product of rings with additional relations

See also: elim_lib; preimage (plural).

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