# Definition:Closure Axioms (Matroid)

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Definition:Closure Operator.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

This page has been identified as a candidate for refactoring of medium complexity.In particular: It is apparent that $\text S 1$ to $\text S 3$ indicate that this is an instance of Definition:Closure Operator, with an extra axiom added on which is less easy to comprehend. It is suggested that some sort of merging may be possible. As it stands, the connection to matroids is not clear, as these axioms are applicable to a general set.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## Definition

Let $S$ be a finite set.

Let $\sigma : \powerset S \to \powerset S$ be a mapping where $\powerset S$ denotes the power set of $S$.

$\sigma$ is said to satisfy the **closure axioms** if and only if:

\((\text S 1)\) | $:$ | \(\ds \forall X \in \powerset S:\) | \(\ds X \subseteq \map \sigma X \) | ||||||

\((\text S 2)\) | $:$ | \(\ds \forall X, Y \in \powerset S:\) | \(\ds X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y \) | ||||||

\((\text S 3)\) | $:$ | \(\ds \forall X \in \powerset S:\) | \(\ds \map \sigma X = \map \sigma {\map \sigma X} \) | ||||||

\((\text S 4)\) | $:$ | \(\ds \forall X \in \powerset S \land x, y \in S:\) | \(\ds y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y} \) |

## Sources

- 1976: Dominic Welsh:
*Matroid Theory*... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid: Theorem $4$