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