# 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 $\mathcal R \subseteq S \times S$ 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 11 subcategories, out of 11 total.

### G

### L

### M

### O

### T

### U

### W

## Pages in category "Total Orderings"

The following 60 pages are in this category, out of 60 total.

### B

### C

### D

### E

- Element of Toset has at most One Immediate Predecessor
- Element of Toset has at most One Immediate Successor
- Elements of Minimal Infinite Successor Set are Well-Ordered
- Equivalence of Definitions of Total Ordering
- Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2
- Extension Theorem for Total Orderings

### F

### I

### 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
- Mind the Gap
- Minimal Element in Toset is Unique and Smallest
- Minimal Infinite Successor 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 Principle

### R

### S

### 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 Sets is Total Ordering
- Unique Isomorphism between Equivalent Finite Totally Ordered Sets
- Upper and Lower Closures of Open Set in GO-Space are Open