Definition:Strict Total Ordering
Definition
Let $\struct {S, \prec}$ be a relational structure.
Let $\prec$ be a strict ordering.
Then $\prec$ is a strict total ordering on $S$ if and only if $\struct {S, \prec}$ has no non-comparable pairs:
- $\forall x, y \in S: x \ne y \implies x \prec y \lor y \prec x$
That is, if and only if $\prec$ is connected.
Also known as
Other terms in use are simple order and order relation.
Some sources, for example 1964: W.E. Deskins: Abstract Algebra and 2000: James R. Munkres: Topology (2nd ed.), call this a linear order.
As this term is also used by other sources to mean total ordering, it is preferred that on $\mathsf{Pr} \infty \mathsf{fWiki}$ the terms partial, total and well, and weak and strict, are the only terms to be used to distinguish between different types of ordering.
Also see
Partial vs. Total Strict Ordering
It is not demanded of a strict ordering $\prec$, defined in its most general form on a set $S$, that every pair of elements of $S$ is related by $\prec$.
They may be, or they may not be, depending on the specific nature of both $S$ and $\prec$.
If it is the case that $\prec$ is a connected relation, that is, that every pair of distinct elements is related by $\prec$, then $\prec$ is called a strict total ordering.
If it is not the case that $\prec$ is connected, then $\prec$ is called a strict partial ordering.
Beware that some sources use the word partial for a strict 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:
- Strict ordering: a strict ordering whose nature (total or partial) is not specified
- Strict partial ordering: a strict ordering which is specifically not total
- Strict total ordering: a strict ordering which is specifically not partial.
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Definition $1.7 \ \text {(a)}$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.18$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations