Definition:Initial Segment
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Definition
Let $\left({S, \preceq}\right)$ be a partially ordered set.
Let $a \in S$.
Then we define:
- $S_a := \left\{{b \in S: b \preceq a \land b \ne a}\right\}$
or alternatively:
- $S_a := \left\{{b \in S: b \prec a}\right\}$
That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$.
$S_a$ is described as the initial segment (of $S$) determined by $a$.
Also known as
The concept of an initial segment is often (and usually more clearly) referred to by its mundane description: the set of preceding elements.
Some sources refer to this concept as a segment.
Some sources reserve the term initial segment exclusively for well-ordered sets, as this is a concept which is usually used in the context of ordinals (which are, by definition, segments of well-ordered sets).
It is also worth noting that the concept of defining the set of all elements which are related to another element crops up throughout the fields of mapping theory and relation theory, defining that set as a segment is usually done only in the context of order theory.
Still other sources write $\mathop{\downarrow} \left({a}\right)$ for $S_a$ and call it the strict lower closure of $a$ (in $S$).
Some sources use $s \left({a}\right)$ for $S_a$.
There is no standard convention for this concept. Therefore it is important, before introducing the notation into a thesis, to define it.
Strict
When it is necessary to distinguish between this and a weak initial segment, this is called a strict initial segment.
Also see
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.7$
