Definition 12.6 Let G = (V,E) be a graph such that . We say that G is a Gröbner graph if G is a directed acyclic graph and for every v V, either T(v,G) is the empty set or v lies in the ideal generated by T(v,G).
Definition 12.7 Let G = (V,E) be a graph and v be a vertex of G. We will say that v is a starting vertex if T(v,G) is the empty set.
The following proposition follows easily from the previous two definitions and mathematical induction. We omit the proof.
Proposition 12.8 Let G = (V,E) be a Gröbner graph, V be a finite set and v be a vertex of G. Let be the set of starting vertices of G. Every element of V lies in the ideal generated by .
Remark 12.9 The hypothesis of Proposition 12.8 can be strengthened. One can replace the hypothesis that V be a finite set with the hypothesis that every element of V has only finitely many ancestors.
If one takes a subgraph of a Gröbner graph G = (V,E) such that
if v is a vertex of the graph and v is not a starting vertex of , thenthen is a Gröbner graph. Other subgraphs of a Gröbner graph may be Gröbner, but the above mentioned subgraphs can be seen to be Gröbner from purely graph-theoretic arguments. We call these subgraphs subGröbner and formalize this with the following definition.
Definition 12.10 A graph G = (V,E) will be called a subGröbner graph with respect to a Gröbner graph if G is a subgraph a and for every v V, either v is a starting vertex of G or T(v,G) = T(v, ).
Note that if one is given a Gröbner graph and a set of vertices , then there is a smallest graph G=(V,E) such that and G is subGröbner with respect to .
If and are graphs, then we let denote the graph and let denote the graph .
Lemma 12.11 Let be a Gröbner graph. If and are subGröbner with respect to , then is subGröbner with respect to and where denotes the set of starting vertices of for any Gröbner graph .
Proof. \ For ease of reference, let , and . We wish to show that . Since , and , it suffices to show that if and only if under the conditions that , or .
If , then if and only if , if and only if and . Therefore, if and only if which occurs if and only if .
If , then if and only if , and . Therefore, if and only if which occurs if and only if .
If , then , if and only if and . Therefore, if and only if which occurs if and only if .
In summary, .
Let v be a vertex of which is not a starting vertex. Therefore, . Without loss of generality, assume that . But then and . Therefore, and so
Therefore, . This completes the proof of Lemma 12.11.
The following lemma follows trivially from the definition of immediate ancestor, the definition of a starting vertex (Definition 12.7) and the definition of subGröbner graph (Definition 12.10).
Lemma 12.12 Let be a Gröbner graph. If = and = are subGröbner with respect to , then is subGröbner with respcet to .
The following technical lemma will be used in the proof of Proposition prop:generating:RR
= (V,E) be a Gröbner graph,
be a finite set,
and for each
if is and element of such that card() card() for each , then is empty for every starting vertex w of .
Proof. \ Suppose, for the purpose of proof by contradiction, that there exists a starting vertex of w of such that is not empty. Since is not empty, there exists which is subGröbner with respect to such that v is a vertex of and every starting vertex of is a member of . By Lemma 12.11, is subGröbner with respect to and . By the choice of , the fact that is a subgraph of and the fact that , the graph equals the graph and so must be a subgraph of . But then, since , w is a starting vertex of and w is a vertex of , w is a member of which is a contradiction. Thus is empty for every starting vertex of w of . This completes the proof of Lemma 12.13.