Category:Equivalence of Definitions of Matroid Circuit Axioms
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This category contains pages concerning Equivalence of Definitions of Matroid Circuit Axioms:
Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
The following definitions for the Matroid Circuit Axioms are equivalent:
Formulation 1
$\mathscr C$ satisfies the circuit axioms:
\((\text C 1)\) | $:$ | \(\ds \O \notin \mathscr C \) | |||||||
\((\text C 2)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \) | ||||||
\((\text C 3)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \) |
Formulation 2
$\mathscr C$ satisfies the circuit axioms:
\((\text C 1)\) | $:$ | \(\ds \O \notin \mathscr C \) | |||||||
\((\text C 2)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \) | ||||||
\((\text C 4)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \) |
Formulation 3
$\mathscr C$ satisfies the circuit axioms:
\((\text C 1)\) | $:$ | \(\ds \O \notin \mathscr C \) | |||||||
\((\text C 2)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \) | ||||||
\((\text C 5)\) | $:$ | \(\ds \forall X \subseteq S \land \forall x \in S:\) | \(\ds \paren {\forall C \in \mathscr C : C \nsubseteq X} \implies \paren {\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \) |
Pages in category "Equivalence of Definitions of Matroid Circuit Axioms"
The following 6 pages are in this category, out of 6 total.
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- Equivalence of Definitions of Matroid Circuit Axioms
- Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2
- Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 3
- Equivalence of Definitions of Matroid Circuit Axioms/Formulation 2 Implies Formulation 1
- Equivalence of Definitions of Matroid Circuit Axioms/Formulation 3 Implies Formulation 1
- Equivalence of Definitions of Matroid Circuit Axioms/Lemma 2