Does anyone know of generalizations of pcf theory where we might consider products of the form:


$$\aleph\_1 \times (\aleph\_2 \times \aleph\_2) \times (\aleph\_3 \times \aleph\_3 \times \aleph\_3) \dots$$


$$(\aleph\_1 \times \aleph\_2 \times \dots) \times (\aleph\_1 \times \aleph\_2 \times \dots) \times \dots$$


or, more abstractly:


$$P\_1 \times P\_2 \times P\_3 \times \dots$$


where each $P\_i$ is a $\mathrm{cof}(P\_i)$-directed partial order. My motivation is that I'm interested in what the relationship between 
$$\max\mathrm{pcf}\langle \aleph\_1,\aleph\_2,\aleph\_2,\aleph\_3,\aleph\_3,\aleph\_3,\dots\rangle$$
and 
$$\max\mathrm{pcf}\langle\aleph\_n\rangle\_{0 < n < \omega }$$
might be because it might help me understand the relationship between 
$$\max \mathrm{pcf} \langle \aleph\_n \rangle\_{0 < n <\omega }$$
and
$$\max \mathrm{pcf} \langle \aleph\_{2n} \rangle\_{0 < n <\omega }$$




---


**Claim**: The $\mathrm{pcf}$ structure on $(\aleph\_1 \times \aleph\_2 \times \dots) \times (\aleph\_1 \times \aleph\_2 \times \dots) \times \dots$ gives nothing new.


**Proof**: For notational convenience, let $A : \omega \to \mathrm{Reg}$ be defined by $A(n) = \aleph\_n$ and let $B : \omega \cdot \omega \to \mathrm{Reg}$ be defined by $B(\omega\cdot m + n) = \aleph\_n$. Define 


$$\mathrm{pcf}(A) = \{\mathrm{cf}(\Pi\_{n<\omega}A(n)/U)\ :\ U \in \beta \omega\}$$
$$\mathrm{pcf}(B) = \{\mathrm{cf}(\Pi\_{\alpha<\omega\cdot\omega}B(\alpha)/U)\ :\ U \in \beta (\omega\cdot\omega)\}$$


Where $\beta X$ denotes the set of all ultrafilters on $X$. It's not hard to see that $\mathrm{pcf}(A) \subseteq \mathrm{pcf}(B)$ and since $\mathrm{pcf}(A)$ is an interval of regular cardinals, it suffices to show that $\max \mathrm{pcf}(B) = \max \mathrm{pcf}(B)$. We know that we can find an everywhere-pointwise-dominating family on $\Pi A$ of size $\lambda := \max\mathrm{pcf}(A)$. If we can find a dominating family on $\Pi B$ of that same size, we'll be done.


So, given a dominating family $\mathcal{F}$ on $\Pi A$ of size $\lambda$, just let $\mathcal{F}^\ast \subseteq \Pi B$ consist of functions of the form:
$$f^\ast (\omega\cdot m + n) = f(n)$$
for each $f \in \mathcal{F}$. Now given $g \in \Pi B$, define:
$$g'(\omega\cdot m + n) = \sup\_{m' \in \omega}g(\omega\cdot m' + n)$$
We see that $g' \geq g$ everywhere pointwise, and $g' \in \Pi B$ since we're always taking countable suprema within uncountable regular cardinals. Since $g'$ has the same value at any of its coordinates that correspond to the same $\aleph\_n$, it's clear that there's some $f \in \mathcal{F}$ such that $f^\ast \in \mathcal{F}^\ast$ dominates $g'$ everywhere, and hence $g$ everywhere.



Generalizations of pcf theory
I don't know if anyone has looked at such things systematically, but I know Shelah has made use of structures of this form at various times. The examples which follow are just what I can remember off-hand; I know there's more buried in his work, but this is where I remembered seeing such a construction:


1. Clause $(\gamma)$ on page 1641 of [Sh:589]:


Applications of PCF theory. J. Symbolic Logic 65 (2000), no. 4, 1624–1674.


(It's available on JSTOR [here](https://www.jstor.org/stable/2695067); the Arxiv version looks like an older iteration)


2. The second proof of his Revised GCH theorem in [Sh:460], in particular Claim 2.6.


The generalized continuum hypothesis revisited. Israel J. Math. 116 (2000), 285–321.


(Available on SpringerLink [here](https://doi.org/10.1007/BF02773223), again the Arxiv version is a bit dated)


I don't think that you get anything really new as far as pcf theory by doing this. Rather, it's just that sometimes such a structure is the right way to organize things.


