Definition of initial segment the definition of initial segment is correct:
"let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in B(a \preceq b \to a \in B )$"
???
Thanks in advance!!

Yes, this is one possible definition of initial segment. Sometimes the term initial segment is defined a little more narrowly; I've seen it used to mean what I would call proper initial segment, i.e., an initial segment by your definition that is not all of $A$, and I've seen it used to mean $\{x\in A:x\preceq a\}$ for some $a\in A$. However, what you've defined is, I think, the most common meaning of the term.
