Quasi o-minimal linear orders

Totally ordered structure $M$ is to be said quasi o-minimal,
if for any $N\equiv M$ its definable subsets are boolean
combination of the $\emptyset$-definable subsets and
intervals with endpoints in $N$.

Let $G=\bigcup_{i<\omega} F_i \bigcup \{\omega, \omega^*,
\omega+\omega^*, \omega^*+\omega,Q\},$
where $F_i$ is finite linear order with $i$ elements, $i<\omega$,
$\ F_0=\emptyset$, $\ Q$ is ordering of rationals.

We consider the following classes of models:
$K_1=\{\sum_{j\in D_1^1}(\sum_{i\in D_2^1} M_{ij}+\sum_{k\in
Q} M'_{kj})\},$
where $D_1^1,D_2^1$ are arbitrary discrete orders and $\forall
j \in D_1^1\ \forall i\in D_2^1\ M_{ij}\equiv N,\ N\in G$
è $\forall k\in Q\ M'_{kj}\equiv N',\ N'\in
\bigcup_{i<\omega}F_i \bigcup \{\omega+\omega^*\}$
$K_l=\{\sum_{j\in D_1^l}(\sum_{i\in D_2^l} M_{ij}+\sum_{k\in
Q} M'_{kj})\},$
where $D_1^l,D_2^l$ are arbitrary discrete orders and $\forall
j \in D_1^l\ \forall i\in D_2^l,\ M_{ij}\equiv N,\ N\in G$
è $\forall k\in Q\ M'_{kj}\equiv N',\ N'\in
\bigcup_{i=1}^{l-1}K'_i,\
K'_i\subset K_i$ è $K'_i$ is subclass of models with both endpoints.\\
{\bf Proposition.}
$\langle M,< \rangle$ is quasi o-minimal model iff
$M\in\bigcup_{i<\omega}K_i.$