Reduced products, true cofinality and scales

We work under The Axiom of Choice in this development. Let $I$ be a set and $J\subseteq \mathcal{P}(I)$ contain $\mathrm{\varnothing }$ and be closed under taking subsets and finite unions. Such a $J$ is an ideal; it is proper if $I\mathrm{\notin }J$. Singletons need not belong to $J$. A set is $J$-positive if it is not in $J$. For positive $B\subseteq I$ write $J\upharpoonright B=J\cap \mathcal{P}(B)$, and write $J+B=\{Y\subseteq I:Y\setminus B\in J\}$ for the ideal obtained by adjoining $B$.

For ordinal-valued functions on $I$, define

$f{\le }_{J}g⟺\{i:f(i)>g(i)\}\in J,f{<}_{J}g⟺\{i:f(i)\ge g(i)\}\in J,$

$f{=}_{J}g⟺\{i:f(i)\mathrm{\ne }g(i)\}\in J\mathrm{.}$

Thus ${<}_J$ means strict inequality outside a small set; it is not defined as ${\le }_J$ together with failure of equality. For the ideal of finite subsets of an infinite set, use ${\le }^{\ast },{<}^{\ast },{=}^{\ast }$ and the word eventually.

For an ordinal function $h:I\to \mathrm{O}\mathrm{r}\mathrm{d}$ the product ${\prod }_{i\in I}h(i)$ consists of functions $f$ with $f(i)<h(i)$ for every $i$. The reduced product modulo $J$ has the ${=}_J$ equivalence classes as elements and the induced comparisons above. Their well-definedness is supplied by the following transfer lemma. If some $h(i)=0$, the product is empty; if $I=\mathrm{\varnothing }$ it contains just the empty function, but there is no proper ideal on $I$.

A family $C$ is cofinal if every product member is ${\le }_J$ some member of $C$. It is strictly cofinal if ${<}_J$ can be used. In the applications $J$ is proper and every $h(i)$ is a nonzero limit ordinal, so each $f$ has a pointwise larger successor function in the product. A scale of length $\lambda$ is a sequence $(f_{\alpha }{)}_{\alpha <\lambda }$ that is strictly ${<}_J$ increasing and cofinal, where $\lambda$ is an infinite regular cardinal in the sense of Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals. If such a sequence exists, its uniquely determined length is the true cofinality, written $\mathrm{tcf}(\prod h\mathrm{/}J)=\lambda$; uniqueness is included in the following lemma. There is no assertion that every reduced product has a scale. Products with a greatest element and improper ideals are not assigned true cofinality by this convention.

A product is $\lambda$-directed when every family of cardinality less than $\lambda$ has a ${\le }_J$ upper bound. If a strict upper bound is required, say strictly $\lambda$-directed. With limit-valued factors the successor function converts the former kind of bound to the latter for proper ideals. For an improper ideal only the weak directedness convention is used.