Category:Reflexive Closures
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This category contains results about Reflexive Closures.
Definition 1
Let $\RR$ be a relation on a set $S$.
The reflexive closure of $\RR$ is denoted $\RR^=$, and is defined as:
- $\RR^= := \RR \cup \set {\tuple {x, x}: x \in S}$
That is:
- $\RR^= := \RR \cup \Delta_S$
where $\Delta_S$ is the diagonal relation on $S$.
Definition 2
Let $\RR$ be a relation on a set $S$.
The reflexive closure of $\RR$ is defined as the smallest reflexive relation on $S$ that contains $\RR$ as a subset.
The reflexive closure of $\RR$ is denoted $\RR^=$.
Definition 3
Let $\RR$ be a relation on a set $S$.
Let $\QQ$ be the set of all reflexive relations on $S$ that contain $\RR$.
The reflexive closure of $\RR$ is denoted $\RR^=$, and is defined as:
- $\RR^= := \bigcap \QQ$
That is:
- $\RR^=$ is the intersection of all reflexive relations on $S$ containing $\RR$.
Also see
Subcategories
This category has only the following subcategory.
Pages in category "Reflexive Closures"
The following 14 pages are in this category, out of 14 total.
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- Reflexive Closure is Closure Operator
- Reflexive Closure is Idempotent
- Reflexive Closure is Inflationary
- Reflexive Closure is Order Preserving
- Reflexive Closure is Reflexive
- Reflexive Closure of Antisymmetric Relation is Antisymmetric
- Reflexive Closure of Relation Compatible with Operation is Compatible
- Reflexive Closure of Strict Ordering is Ordering
- Reflexive Closure of Strict Total Ordering is Total Ordering
- Reflexive Closure of Transitive Antisymmetric Relation is Ordering
- Reflexive Closure of Transitive Relation is Transitive