# 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 8 subcategories, out of 8 total.

### G

### L

### M

### O

### T

### U

### W

## Pages in category "Total Orderings"

The following 54 pages are in this category, out of 54 total.

### 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 from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication
- Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 1
- Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 2
- 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
- Ordered Product of Tosets is Totally Ordered Set
- Ordered Product of Tosets is Totally Ordered Set/General Result
- Ordered Sum of Tosets is Totally Ordered Set
- Ordered Sum of Tosets is Totally Ordered Set/General Result