Definition of "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.

1999, Peter J. Cameron, Permutation Groups, Cambridge University Press, page 124

We say a set S ⊂ V {\displaystyle S\subset V} is irredundant if for any v ∈ S {\displaystyle v\in S} there exists a vertex u ∈ V {\displaystyle u\in V} such that v {\displaystyle v} dominates u {\displaystyle u} and S ∖ { v } {\displaystyle S\setminus \left\{v\right\}} does not dominate u {\displaystyle u} . We call any such vertex u {\displaystyle u} a private vertex for v {\displaystyle 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 {\displaystyle G} as in Figure 1.

2010, Marek Cygan, Marcin Pilipczuk, Jakub Onufry Wojtaszczyk, “Irredundant Set Faster than O(2n)”, in Josep Diaz, Tiziana Calamoneri, editors, Algorithms and Complexity: 7th International Conference, CIAC 2010, Proceedings, Springer,, page 289

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.

2013, Donald D. Givone, Digital Principles and Design, McGraw-Hill, page 178