Definition:End-Extension of Ordered Set
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Definition
Let $\struct {S, \preccurlyeq_S}$ and $\struct {T, \preccurlyeq_T}$ be ordered sets such that $S \subseteq T$.
Let ${\preccurlyeq_S} = {\preccurlyeq_T} {\restriction_S}$ be the restriction of $\preccurlyeq_T$ to $S$.
Let $i_S: S \to T$ denote the inclusion mapping from $S$ to $T$:
- $\forall s \in S: \map {i_S} s = s$
Let:
- $\forall s \in S: \forall t \in T \setminus S: x \prec_T y$
Then $\struct {T, \preccurlyeq_T}$ is an end-extension of $\struct {S, \preccurlyeq_S}$.
Examples
Closed Real Interval
The closed real interval $\closedint 0 1$ is:
- an end-extension of the half-open interval $\hointr 0 1$
- but not an end-extension of the open interval $\openint 0 1$.
Also see
- Results about end-extensions can be found here.
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations