Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1
Jump to navigation
Jump to search
Definition
Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
$\mathscr C$ is said to satisfy the circuit axioms if and only if:
\((C1)\) | $:$ | \(\displaystyle \O \notin \mathscr C \) | ||||||
\((C2)\) | $:$ | \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) | \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \) | |||||
\((C3)\) | $:$ | \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) | \(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \) |
Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid, Theorem 5