# Definition:Ordering/Definition 1

## Definition

Let $S$ be a set.

An **ordering on $S$** is a relation $\mathcal R$ on $S$ such that:

\((1)\) | $:$ | $\mathcal R$ is reflexive | \(\displaystyle \forall a \in S:\) | \(\displaystyle a \mathop {\mathcal R} a \) | ||||

\((2)\) | $:$ | $\mathcal R$ is transitive | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle a \mathop {\mathcal R} b \land b \mathop {\mathcal R} c \implies a \mathop {\mathcal R} c \) | ||||

\((3)\) | $:$ | $\mathcal R$ is antisymmetric | \(\displaystyle \forall a \in S:\) | \(\displaystyle a \mathop {\mathcal R} b \land b \mathop {\mathcal R} a \implies a = b \) |

## Notation

Symbols used to denote a general ordering relation are usually variants on $\preceq$, $\le$ and so on.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general ordering relation it is recommended to use $\preceq$ and its variants:

- $\preccurlyeq$
- $\curlyeqprec$

To denote the conventional ordering relation in the context of numbers, the symbol $\le$ is to be used, or its variants:

- $\leqslant$
- $\leqq$
- $\eqslantless$

The symbol $\subseteq$ is universally reserved for the subset relation.

- $a \preceq b$

can be read as:

**$a$ precedes, or is the same as, $b$**.

Similarly:

- $a \preceq b$

can be read as:

**$b$ succeeds, or is the same as, $a$**.

If, for two elements $a, b \in S$, it is not the case that $a \preceq b$, then the symbols $a \npreceq b$ and $b \nsucceq a$ can be used.

## Partial vs. Total Ordering

It is not demanded of an ordering $\preceq$, defined in its most general form on a set $S$, that *every* pair of elements of $S$ is related by $\preceq$. They may be, or they may not be, depending on the specific nature of both $S$ and $\preceq$.

If it *is* the case that $\preceq$ is a connected relation, that is, that every pair of distinct elements is related by $\preceq$, then $\preceq$ is called a total ordering.

If it is *not* the case that $\preceq$ is connected, then $\preceq$ is called a partial ordering.

Beware that some sources use the word **partial** for an ordering which **may or may not** be connected, while others insist on reserving the word **partial** for one which is specifically **not** connected.

It is wise to be certain of what is meant.

As a consequence, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we resolve any ambiguity by reserving the terms for the objects in question as follows:

**Partial ordering**: an ordering which is specifically**not**total

**Total ordering**: an ordering which is specifically**not**partial.

## Also see

- Results about
**orderings**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$: Definition $1.7 \ \text {(b)}$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.1$: Partially ordered sets - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 3$: Equivalence relations and quotient sets: Binary relations - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Order