Order Theory

some necessary definitions:

partially ordered set

A partially ordered set (or poset) is a set equipped with a binary relation: $(S,⊑)$, such that the relation is:

• reflexive
• transitive
• anti-symmetric

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least pre-fixed point

a pre-fixed point of a function $f:D\to D$, where $D$ is a poset, is some $d\in D$ such that $f(d)⊑d$.

the least pre-fixed point $\text{fix}(f)$ is the least such point. that is: $\forall d\in D$, if $d$ is a pre-fixed point then $\text{fix}(f)⊑d$.

notably, $\text{fix}(f)$ is also the least fixed point of $f$, since all fixed points are also pre-fixed points.

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chain-complete poset

a poset where every countable, increasing chain has a least upper bound

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in DenSem we take chain to mean a countable, increasing chain of elements in a poset.

domain

a chain-complete poset with a least element

this least element is typically denoted $\perp$.

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admissible subset

A subset of a domain $D$ that is chain-closed and contains $\perp$

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