Axiom:Base Axiom (Matroid)/Formulation 1
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Definition
Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
$\mathscr B$ is said to satisfy the base axiom if and only if:
\((\text B 1)\) | $:$ | \(\ds \forall B_1, B_2 \in \mathscr B:\) | \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y \in \mathscr B \) |
See also
Sources
- 2011: James Oxley: Matroid Theory (2nd ed.) Chapter $11$ Submodular Function and Matroid Union $\S 11.3$ Matroid union and its applications, Exercise $10$
- 2018: Bernhard H. Korte and Jens Vygen: Combinatorial Optimization: Theory and Algorithms (6th ed.) Chapter $13$ Matroids $\S 13.2$ Other Matroid Axioms, Theorem $13.9$