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 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 in particular, it is referred to as (strict) lower closure.

In such a context, the notation $a^\prec$ is used on to denote the initial segment.

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