Definition:Increasing Mappings Satisfying Inclusion in Lower Closure

Definition
Let $R = \struct {S, \preceq}$ be an ordered set.

Let $\map {\it Ids} R$ be the set of all ideals in $R$.

Let $L = \struct {\map {\it Ids} R, \precsim}$ be an ordered set where $\precsim \mathop = \subseteq \restriction_{\map {\it Ids} R \times \map {\it Ids} R}$

Then the ordered set $M$ of increasing mappings $f: R \to L$ satisfying $\forall x \in S: \map f x \subseteq x^\preceq$

is defined by:
 * $M = \struct {F, \preccurlyeq}$

where:
 * $F = \leftset {f: S \to \map {\it Ids} R: f}$ is increasing mapping $\rightset {\land \forall x \in S: \map f x \subseteq x^\preceq}$

and:
 * $\preccurlyeq$ is ordering on mappings generated by $\precsim$

where $x^\preceq$ denotes the lower closure of $x$.

Also see

 * Correctness of Definition of Increasing Mappings Satisfying Inclusion in Lower Closure