Ground Set (Matroid)

discrete-mathematics combinatorics

Definition

Ground Set (Matroid)

In a matroid $M=(E,\mathcal{I})$, the ground set is the finite non-empty set $E$ whose subsets are classified as independent or dependent by the family $\mathcal{I}$. It is the universe of objects the matroid reasons about — the raw elements from which independent sets are drawn.

The family $\mathcal{I}\subseteq 2^E$ is the collection of subsets of $E$ declared independent. It is the rule that decides, for each subset of the ground set, whether it counts as independent or not. The matroid structure is determined not by $E$ alone, but by the choice of $\mathcal{I}$ on it: the same ground set can carry different matroids.