# Category:Total Orderings

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This category contains results about **Total Orderings**.

Definitions specific to this category can be found in Definitions/Total Orderings.

Let $\RR \subseteq S \times S$ be a relation on a set $S$.

$\RR$ is a **total ordering** on $S$ if and only if:

That is, $\RR$ is an ordering with no non-comparable pairs:

- $\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$

## Subcategories

This category has the following 13 subcategories, out of 13 total.

## Pages in category "Total Orderings"

The following 71 pages are in this category, out of 71 total.

### B

### C

- Cardinals are Totally Ordered
- Characteristic of Increasing Mapping from Toset to Order Complete Toset
- Complement of Strict Total Ordering
- Condition for Ordered Set of All Mappings to be Total Ordering
- Condition for Power Set to be Totally Ordered
- Condition for Total Ordering to be Well-Ordering
- Condition for Uniqueness of Increasing Mappings between Tosets

### D

### E

### F

### I

- Immediate Predecessor under Total Ordering is Unique
- Immediate Successor is Unique in Toset
- Immediate Successor under Total Ordering is Unique
- Injection Induces Total Ordering
- Intersection of Strict Lower Closures in Toset
- Intersection of Strict Upper Closures in Toset
- Intersection of Weak Lower Closures in Toset
- Intersection of Weak Upper Closures in Toset
- Isomorphism Class of Total Orderings

### L

### M

- Mapping from Totally Ordered Set is Dual Order Embedding iff Strictly Decreasing
- Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing
- Mapping on Total Ordering reflects Transitivity
- Maximal Element in Toset is Unique and Greatest
- Minimal Element in Toset is Unique and Smallest
- Minimally Inductive Set is Well-Ordered
- Monomorphism from Total Ordering

### O

- Open Ray is Dual to Open Ray
- Order Embedding between Quotient Fields is Unique
- Order Isomorphism between Tosets is not necessarily Unique
- Order Isomorphism on Totally Ordered Set preserves Total Ordering
- Order Sum of Totally Ordered Sets is Totally Ordered
- Ordered Sum of Tosets is Totally Ordered Set
- Ordered Sum of Tosets is Totally Ordered Set/General Result
- Ordering on Singleton is Total Ordering
- Ordering Principle

### R

### S

- Simple Order Product of Pair of Totally Ordered Sets is Total iff One Factor is Singleton
- Simple Order Product of Totally Ordered Sets may not be Totally Ordered
- Strictly Order-Preserving and Order-Reversing Mapping on Strictly Totally Ordered Set is Injection
- Subset of Join Semilattice on Total Ordering is Closed
- Subset of Toset is Toset
- Surjection on Total Ordering reflects Preordering

### T

### U

- Union of Overlapping Convex Sets in Toset is Convex
- Union of Overlapping Convex Sets in Toset is Convex/Infinite Union
- Union of Total Ordering with Lower Sections is Total Ordering
- Unique Isomorphism between Equivalent Finite Totally Ordered Sets
- Upper and Lower Closures of Open Set in GO-Space are Open
- Upper Closure is Strict Upper Closure of Immediate Predecessor