Irredundant

Theorem 4.23 The following conditions on a finite permutation group are equivalent: (a) all irredundant bases have the same size; (b) the irredundant bases are invariant under re-ordering; (c) the irredundant bases are the bases of a matroid.

We say a set S#92;subsetV is irredundant if for any v#92;inS there exists a vertex u#92;inV such that v dominates u and S#92;setminus#92;left#92;#123;v#92;right#92;#125; does not dominate u. We call any such vertex u a private vertex for v. An irredundant set is called inclusion–maximal if it is not a proper subset of any other irredundant set. Note that an inclusion–maximal irredundant set does not necessarily have to dominate the whole vertex set of G as in Figure 1.

If each of the 10 irredundant expressions is now evaluated by the cost criterion proposed in Sec. 4. 1 involving the total number of gate inputs. then the minimal sums are obtained since a minimal expression is irredundant.