Category:Minimally Closed Classes under Progressing Mapping

From ProofWiki
Jump to navigation Jump to search

This category contains pages concerning Minimally Closed Class under Progressing Mapping:


Statement of Conditions:

Let $N$ be a class which is closed under a progressing mapping $g$.

Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.


Then the following results hold:


Minimally Closed Class under Progressing Mapping induces Nest

For all $x, y \in N$:

either $\map g x \subseteq y$ or $y \subseteq x$

and $N$ forms a nest:

$\forall x, y \in N: x \subseteq y$ or $y \subseteq x$


Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element

Every bounded subset of $N$ has a greatest element.


Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element

$g$ has no fixed point, unless possibly the greatest element, if there is one.


Minimally Closed Class under Progressing Mapping is Well-Ordered

$N$ is well-ordered under the subset relation.


Smallest Element of Minimally Closed Class under Progressing Mapping

$b$ is the smallest element of $N$.