\begin{definition}[Definition:Increasing Mappings Satisfying Inclusion in Lower Closure]
Let $R = \left({S, \preceq}\right)$ be an ordered set.
Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$.
Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set
where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$
Then '''ordered set $M$ of increasing mappings $f:R \to L$ satisfying''' $\forall x \in S: f\left({x}\right) \subseteq x^\preceq$
is defined by
:$M = \left({F, \preccurlyeq}\right)$
where
:$F = \left\{ {f: S \to {\it Ids}\left({R}\right): f}\right.$ is increasing mapping $\left.{\land \forall x \in S: f\left({x}\right) \subseteq x^\preceq}\right\}$
and
:$\preccurlyeq$ is ordering on mappings generated by $\precsim$
where $x^\preceq$ denotes the lower closure of $x$.
\end{definition}