Definition:Initial Segment/Also known as

Definition

The concept of an initial segment is often (and usually more clearly) referred to by its mundane description: the set of strictly preceding elements.

Some sources refer to this concept as a segment.

Some sources refer to this concept as a section.

When it is necessary to distinguish between this and a weak initial segment, this is called a strict initial segment.

Some sources use the term strict lower closure, but $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve that term for when the ordering in question is not necessarily a well-ordering.

There is no standard notation or convention for this concept. Therefore it is important, before introducing the notation into a thesis, to define it.

In the context of a general ordered set, the concept is used more broadly, and there are far too many synonyms for this and related concepts.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ in particular, it is referred to as (strict) lower closure.

In such a context, the notation $a^\prec$ is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the initial segment.

Some sources use $\map s a$ for $S_a$.