The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the relevant properties of these posets to be abstracted. I'm wondering if this has been done before. As evidence that it's at least possible, here's one possible approach.
The Setting:
Definition: Let $\kappa$ be an uncountable regular cardinal. Say that a poset $P$ is $\kappa$-generated by $A \subseteq P$ if the following conditions hold:
- $P$ is directed and has $\kappa$-small directed joins.
- Each $a \in A$ is compact in $P$, i.e. $a \leq \vee_i p_i \Rightarrow \exists i\, a \leq p_i$ if the join is directed.
- For each $p \in P$, the set $A_{\leq p} := \{a \in A \mid a \leq p\}$ is $\kappa$-small.
- For each $p \in P$, the join $\vee A_{\leq p}$ exists in $P$ and is equal to $p$.
Examples: Let $\kappa$ be an uncountable regular cardinal.
- The wellorder $\kappa$ is $\kappa$-generated by itself.
- For any set $A$, the restricted powerset $P_\kappa(A)$ is $\kappa$-generated by $A$.
- For any join-semilattice $A$, the $\kappa$-small Ind-completion of $A$ is $\kappa$-generated by $A$.
Some familiar concepts:
Definition: Let $\kappa$ be an uncountable regular cardinal and let $P$ be a poset $\kappa$-generated by $A$.
- A subset $S \subseteq P$ is club if it is closed under $\kappa$-small directed colimits and every $p \in P$ has an upper bound in $S$, and stationary if it meets every club set nontrivially.
- A filter $\mathcal F \subseteq P(P)$ is fine if for every $a \in A$ it contains the set $P_{\geq a} := \{p \in P \mid p \geq a\}$.
- A filter $\mathcal F \subseteq P(P)$ is normal if for every $A$-indexed family $(X_a)_{a \in A}$ with $X_a \in \mathcal F$, the diagonal intersection $\Delta_{a \in A} X_a := \{p \in P \mid p \in \cap_{a \leq p} X_a\}$ is in $\mathcal F$.
Observation: If $\mathcal F$ is fine and $\kappa$-complete, then $\mathcal F$ contains each $P_{\geq p}$ for $p \in P$.
Some familiar theorems:
In this setting, we can recover some basic results about stationarity. Applied to the examples (1) and (2) above, one gets some familiar facts.
Proposition: Let $\kappa$ be an uncountable regular cardinal and $P$ a poset $\kappa$-generated by $A$.
- The club sets generate the minimal $\kappa$-complete fine normal filter $\mathcal F_{club} \subseteq P(P)$.
- Fodor's lemma holds: if $f: P \to A$ is regressive in that $f(s) \leq s$ for each $s \in S$ where $S \subseteq P$ is stationary, then there is a stationary $S_0 \subseteq S$ such that $f|_{S_0}$ is constant.
- The club filter is generated by the sets $Cl_f = \{p \in P \mid \forall a \in b,\, f(b) \leq p\}$ for $f: P_\omega(A) \to P$.
Questions:
- Are the notions of clubness / stationarity / fineness / normality / diagonal intersection with respect to a poset in the literature somewhere?
- If so, what are they used for?
- Is Fodor's lemma known in some form for some general class of posets?
