Definition:Strict Ordering/Notation
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Definition
Symbols used to denote a general strict ordering are usually variants on $\prec$, $<$ and so on.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general strict ordering it is recommended to use $\prec$.
To denote the conventional strict ordering in the context of numbers, the symbol $<$ is to be used.
The symbol $\subset$ is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is ambiguous in the literature, and this can be a cause of confusion and conflict.
Hence the symbols $\subsetneq$ and $\subsetneqq$ are used for the (proper) subset relation.
\(\ds a\) | \(\prec\) | \(\ds b\) | can be read as: | \(\quad\) $a$ (strictly) precedes $b$ | ||||||||||
\(\ds a\) | \(\prec\) | \(\ds b\) | can also be read as: | \(\quad\) $b$ (strictly) succeeds $a$ |
If, for two elements $a, b \in S$, it is not the case that $a \prec b$, then the symbols $a \nprec b$ and $b \nsucc a$ can be used.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering