Definition:Strict Lower Closure/Element/Also known as

Strict Lower Closure of Element: Also known as

The strict lower closure of an element $a$ also goes by the names:

strict down-set

strict down set

initial segment (particularly when $\preccurlyeq$ is a well-ordering)

strict initial segment

set of (strictly) preceding elements to $a$

The term (strict) initial segment is usually seen in discussion of the properties of ordinals.

In this context, the notation $S_a$ or $\map s a$ can often be found for $a \in S$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term an initial segment of $S$ is specifically reserved for the strict lower closure of some element $a$ of $S$ under a well-ordering.

In particular, see Initial Segment of Natural Numbers.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries