Category:Lower Closures
Jump to navigation
Jump to search
This category contains results about Lower Closures in the context of Order Theory.
Definitions specific to this category can be found in Definitions/Lower Closures.
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $a \in S$.
The lower closure of $a$ (in $S$) is defined as:
- $a^\preccurlyeq := \set {b \in S: b \preccurlyeq a}$
That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
L
Pages in category "Lower Closures"
The following 24 pages are in this category, out of 24 total.
L
- Lower Closure is Closure Operator
- Lower Closure is Dual to Upper Closure
- Lower Closure is Increasing
- Lower Closure is Lower Section
- Lower Closure is Strict Lower Closure of Immediate Successor
- Lower Closure of Element is Ideal
- Lower Closure of Singleton
- Lower Closure of Subset is Subset of Lower Closure
- Lower Closures are Equal implies Elements are Equal
S
- Segment of Auxiliary Relation is Subset of Lower Closure
- Strict Lower Closure in Restricted Ordering
- Strict Lower Closure is Dual to Strict Upper Closure
- Strict Lower Closure is Lower Section
- Strict Lower Closure of Element is Proper Lower Section
- Strict Lower Closure of G-Tower is Set of Elements which are Proper Subsets
- Strict Lower Closure of Limit Element is Infinite
- Supremum of Lower Closure of Element
- Supremum of Lower Closure of Set