Hello, I am recently in this new topic and I am not sure how to interpret the initial segment and chain definitions, someone could clarify my doubt with this example. why is it chain? why is it initial segment?
The figure represents a partially ordered class with six elements. Note that $ \{a, b, c\} $ and $ \{d, e, b, c\}$ are chains of $A$; $S_e = \{b, c, f \} $ is the initial segment determined by $e$; if $L =\{a, b, c\}$ and $U =\{d, e, f\}$, then $\{L, U\}$ is a cut.
Definition Let $A$ be a partially ordered class and suppose $a ∈ A$. The initial segment of $A$ determined by $a$ is the class $S_a$, defined as follows:
$S_a= \{x \in A: x \prec a \} $