Category:Definitions/Initial Segments

This category contains definitions related to Initial Segments.
Related results can be found in Category:Initial Segments.

Let $\struct {S, \preceq}$ be a well-ordered set.

Let $a \in S$.

The initial segment (of $S$) determined by $a$ is defined as:

$S_a := \set {b \in S: b \preceq a \land b \ne a}$

which can also be rendered as:

$S_a := \set {b \in S: b \prec a}$

That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$.

That is, $S_a$ is the strict lower closure of $a$ (in $S$).

By extension, $S_a$ is described as an initial segment (of $S$).