Singular Manual: Groebner bases in G-algebras

Forward: Syzygies and resolutions (plural)

7.4.2 Groebner bases in G-algebras

We follow the notations, used in the SINGULAR Manual (e.g. in Standard bases).

For a -algebra , we denote by ${}_{A} \langle g_1, \dots, g_s \rangle$ the left submodule of a free module , generated by elements $\{g_1, \dots, g_s\}\subset A^r$ .

Let be a fixed monomial well-ordering on the -algebra with the Poincar@'e-Birkhoff-Witt (PBW) basis $\{x^{\alpha} = x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n\}$ . For a given free module with the basis $\{e_1,\ldots, e_r\}$ , denotes also a fixed module ordering on the set of monomials $\{x^{\alpha} e_i \mid \alpha\in {\bf N}^n, 1\leq i\leq r \}$ .

Definition

For a set $S \subset A^r$ , define

to be the

-vector space, spanned on the leading monomials of elements of

, $L(S) = \oplus \{K x^{\alpha} e_i \mid \exists s \in S, \hbox{LM(s)}=x^{\alpha}e_i\}$ .

We call the span of leading monomials of .

Let $I \subset A^r$ be a left -submodule. A finite set $G\subset I$ is called a left Groebner basis of if and only if , that is for any $f \in I\setminus \{ 0 \}$ there exists a $g\in G$ satisfying $\hbox{LM}(g) \mid \hbox{LM}(f)$ , i.e., if $\hbox{LM}(f) = x^{\alpha}e_i$ , then $\hbox{LM}(f) = x^{\beta}e_i$ with $\beta_j \leq \alpha_j, \; 1\leq j \leq n$ .

Remark: In general non-commutative algorithms are working with global well-orderings only (see PLURAL, Monomial orderings and Term orderings), unless we deal with graded commutative algebras via Graded commutative algebras (SCA).

A Groebner basis $G\subset A^r$ is called minimal (or reduced) if $0\notin G$ and if $\hbox{LM}(g)\notin L(G\setminus \{ g \})$ for all $g\in G$ . Note, that any Groebner basis can be made minimal by deleting successively those with $\hbox{LM}(h)\mid \hbox{LM}(g)$ for some $h\in G\setminus\{g \}$ .

For $f\in A^r$ and $G\subset A^r$ we say that is completely reduced with respect to if no monomial of is contained in .

Left Normal Form

A map $\hbox{NF} : A^r \times \{G \mid G\ \hbox{ a (left) Groebner basis}\} \to A^r, (f\vert G) \mapsto \hbox{NF}(f\vert G)$ , is called a (left) normal form on if for any $f\in A^r$ and any left Groebner basis the following holds:

(i) $\hbox{NF}(0\vert G) = 0$ ,

(ii) if $\hbox{NF}(f\vert G) \not= 0$ then $\hbox{LM}(g)$ does not divide $\hbox{LM}(\hbox{NF}(f\vert G))$ for all $g\in G$ ,

(iii) $f - \hbox{NF}(f\vert G)\in {}_{A}\langle G \rangle$ .

$\hbox{NF}(f\vert G)$ is called a left normal form of with respect to (note that such a map is not unique).

Remark: As we have already mentioned in the definitions ideal and module (see PLURAL), by NF (or reduce) PLURAL understands a left normal form. Note, that rightNF from nctools_lib allows to compute a right normal form.

Left ideal membership (plural)

For a left Groebner basis of the following holds: $f \in I$ if and only if the left normal form $\hbox{NF}(f\vert G) = 0$ .

For computing a left Groebner basis G of I, use std (plural).

For computing a left normal form of f with respect to G, use reduce (plural).

Right ideal membership (plural)

The right ideal membership is analogous to the left one:

for computing a right Groebner basis G of I, use rightstd (letterplace) from nctools_lib,

for computing a right normal form of f with respect to G, use rightNF from nctools_lib.

Two-sided ideal membership (plural)

Let be a two-sided ideal and be a two-sided Groebner basis of .

Then $f \in J$ if and only if the left normal form $\hbox{NF}(f\vert T) = 0$ .

For computing a two-sided Groebner basis T of J, use twostd (plural),

for computing a normal form of f with respect to T, use reduce (plural).

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