Initial segment of a well-order

Let $(W,<)$ be a well-order (Well-order and well-ordered set).

A subset $I\subseteq W$ is an initial segment of $W$ when it is downward closed: if $x\in I$ and $y<x$ then $y\in I$. It is a proper initial segment when $I\mathrm{\ne }W$.

For $a\in W$ write

$W_{<a}:=\{x\in W:x<a\},W_{\le a}:=\{x\in W:x\le a\}=W_{<a}\cup \{a\},$

and call $W_{<a}$ the initial segment determined by $a$.

Every initial segment $I$, carrying the order inherited from $W$, is itself a well-order: the inherited order is total, and a nonempty subset of $I$ is a nonempty subset of $W$, so it has a least element, which lies in $I$.

• $\mathrm{\varnothing }$ and $W$ are initial segments of $W$; each $W_{<a}$ is a proper initial segment, since $a\mathrm{\notin }{W}_{<a}$ by irreflexivity; and each $W_{\le a}$ is an initial segment.
• Every proper initial segment is $W_{<a}$ for exactly one $a$. Let $I⊊W$ be an initial segment and put $a=\min (W\setminus I)$, which exists because $W\setminus I$ is a nonempty subset of the well-order $W$ (Well-order and well-ordered set). If $x<a$ then $x\mathrm{\notin }W\setminus I$ by minimality of $a$, so $x\in I$; hence $W_{<a}\subseteq I$. Conversely let $x\in I$. Then $x\mathrm{\ne }a$, because $a\mathrm{\notin }I$, and $a<x$ is impossible, because downward closure would then put $a\in I$; so $x<a$ by trichotomy, and $I\subseteq W_{<a}$. Therefore $I=W_{<a}$. For uniqueness, suppose $W_{<a}=W_{<b}$ with $a\mathrm{\ne }b$, say $a<b$; then $a\in W_{<b}=W_{<a}$, that is $a<a$, which is impossible.
• Nesting. If $b<a$ then $(W_{<a}{)}_{<b}=W_{<b}$, so an initial segment of an initial segment of $W$ is an initial segment of $W$. This is used whenever two well-orders are compared.
• The initial segments of $W$ are therefore exactly the sets $W_{<a}$ for $a\in W$, together with $W$ itself, and inclusion orders them in the same shape as $W$ with one extra element added on top.
• The corresponding notion for a general poset would be a downward closed set, or "lower set". Nothing on this page needs it outside the well-ordered case, so the definition is stated only there, where the second clause above makes the family of initial segments completely explicit.