I am going over some notes and am a little confused over a couple details. What I have:
Let $(X,\leqslant_X)$ and $(S,\leqslant_S)$ be binary structures such that:
- $S\subseteq X$
- $\color{blue}{\leqslant_S\;\subseteq\; \leqslant_X}$
- $\forall s\in S,\forall x\in X\;(x\,{\color{red} \leqslant}\, s\implies x\in S)$
Then $(S,\leqslant_S)$ is called an initial segment of $(X,\leqslant_X)$.
The points I am a little unsure of:
- in the expression in blue, are we allowing $\leqslant_S$ to not be the restriction of $\leqslant_X$ to $S$ but to be a proper subset of this restriction? I don't quite understand the need for this line otherwise.
- the symbol in red, should I take this to mean $\leqslant_X$? Again, I'm confused as to why we don't write the relation as $\leqslant$ throughout, and perhaps write $\leqslant\big|_S$ for the restriction on the initial segment.
- finally, the $\leqslant$ symbol is suggestive of at least a pre-order, is the concept of an initial segment of not much interest for arbitrary binary relations? The definition above looks like it would apply to any such relation.
This has been made
unduely complicated and myopic.
Let S be an ordered set.
A subset S is an intial segment of S
when A is a lower set:
for all s in S, a in A, (s <= a implies s in A).
Viewed as an ordered set in itself,
A has the inherited or restricted order of S.