| 1. | A finitary closure operator with this property is called a matroid.
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| 2. | The Galois connection is not uniquely determined by the closure operator.
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| 3. | For more on this matter, see closure operator below.
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| 4. | Finitary closure operators with this property give rise to antimatroids.
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| 5. | A closure operator on a partially ordered set is determined by its closed elements.
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| 6. | It is an example of a closure operator; all closure operators are idempotent functions.
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| 7. | It is an example of a closure operator; all closure operators are idempotent functions.
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| 8. | Then that closure operator can be shown to satisfy the axioms of a preclosure operator.
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| 9. | This allows to define new local structures on the category ( such as a closure operator ).
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| 10. | In fact, " every " closure operator arises in this way from a suitable Galois connection.
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