Towers of singular order type in pseudo-tree algebras: extending Monk's regular-cardinal characterization

Monk (2009) characterized, for each regular cardinal $\kappa$, exactly when the pseudo-tree algebra $\operatorname{Treealg}(T)$ has a tower of order type $\kappa$ — a strictly increasing $\kappa$-sequence of nonunit elements with join $1$ — via an intricate twelve-condition tree-theoretic equivalence, and proved that every nonempty set of regular cardinals occurs as the tower spectrum of an atomless tree algebra. The singular case is entirely untreated: the source's definition of tower presumes regularity, and the weak-tower spectrum problems that Monk's 2014 monograph poses for arbitrary Boolean algebras (its Problems 32–33) remain open, with no pseudo-tree special case known. This record asks for the singular-cardinal completion of the theory: necessary and sufficient conditions on $T$ for $\operatorname{Treealg}(T)$ to have a weak tower of a prescribed singular order type, and a characterization of the weak-tower spectra of pseudo-tree and tree algebras.

Background

Towers are among the standard small cardinal invariants of a Boolean algebra (BA). A BA here is an algebra of sets, and a tower in a BA $A$ is a strictly increasing sequence $\langle a_\alpha:\alpha<\kappa\rangle$ of elements of $A\setminus\{1\}$ whose join is $1$, the top element; classically $\kappa$ is required to be an infinite regular cardinal, the least such $\kappa$ is denoted $\operatorname{tow}(A)$, and the tower spectrum $\operatorname{towspect}(A)$ is the set of regular $\kappa$ for which $A$ has a tower of length $\kappa$. A pseudo-tree is a partially ordered set $T$ in which every initial segment $T\downarrow t=\{s\in T:s<t\}$ is linearly ordered; if each $T\downarrow t$ is well-ordered, then $T$ is a tree. For a pseudo-tree $T$, $\operatorname{Treealg}(T)$ denotes the BA of subsets of $T$ generated by the principal up-sets $T\uparrow t=\{s\in T:t\le s\}$. These pseudo-tree algebras, introduced by Koppelberg and Monk and developed further by Bekkali, are exactly the subalgebras of interval algebras $\operatorname{Intalg}(L)$ (the BAs of finite unions of half-open intervals of a linear order $L$) by a theorem of Heindorf, so they simultaneously generalize interval algebras and tree algebras. Every atomless BA has a tower, yet tower existence can fail in general: there are explicit trees, and explicit linear orders, whose associated algebras have none. Monk (2007) characterized when an interval algebra has a tower of a given regular order type and showed that every nonempty set of regular cardinals is the tower spectrum of some atomless BA. Monk (2009) then settled the pseudo-tree case: for a regular cardinal $\kappa$ fixed in advance, he gave a necessary-and-sufficient characterization of when $\operatorname{Treealg}(T)$ has a tower of order type $\kappa$, in the form of a long disjunction of twelve tree-theoretic conditions (I)–(XII) concerning non-decreasing $\kappa$-sequences in $T$, sets of incomparable elements, and countable decreasing sequences above them. The corollary for trees reads: $\operatorname{Treealg}(T)$ has a tower iff $T$ has an element with exactly $\omega$ immediate successors, or a chain of countable limit length with at most $\omega$ immediate successors, or a chain of some uncountable regular order type with only finitely many immediate successors. Monk also proved, using trees of very large branching (and, for the interval-algebra version, Hausdorff's classical constructions of dense linear orders with prescribed gap characters), that every nonempty set of regular cardinals is $\operatorname{towspect}(A)$ for some atomless tree algebra $A$, and he gave a clean equivalent condition for limit-normal trees (every initial chain of limit ordinal length has at most one immediate successor). This entire theory, like the definition of tower it uses, is confined to regular $\kappa$. In his 2014 monograph, Monk introduces the corresponding unrestrained notions — a weak tower allows any infinite cardinal length $\kappa$, a very weak tower any limit ordinal length — records only the elementary fact that weak-tower order types are closed downward inside each cofinality class, and poses the weak- and very-weak-tower spectrum problems for arbitrary atomless BAs as his Problems 32 and 33, left open; the monograph's tree-algebra chapter still treats only regular $\kappa$, referring back to the 2009 paper for pseudo-trees. Thus for pseudo-tree algebras nothing is known about towers of singular order type: which singular lengths occur, and under what tree-theoretic conditions on $T$.

Problem Statement

Extend Monk's 2009 tower-existence theory for pseudo-tree algebras from regular to singular cardinals. Precisely: (1) Characterization. For each singular cardinal $\kappa$, find necessary and sufficient conditions on a pseudo-tree $T$ under which $\operatorname{Treealg}(T)$ has a weak tower of order type $\kappa$, i.e. a strictly increasing sequence $\langle a_\alpha:\alpha<\kappa\rangle$ of elements of $\operatorname{Treealg}(T)\setminus\{1\}$ with $\sum_{\alpha<\kappa}a_\alpha=1$, in the tree-theoretic spirit of the 2009 regular-cardinal theorem; equivalently, extend that theorem from regular $\kappa$ to arbitrary infinite cardinals. (2) Spectrum. Determine which sets of cardinals can arise as $\{\kappa:\operatorname{Treealg}(T)\text{ has a weak tower of order type }\kappa\}$ when $T$ ranges over pseudo-trees, respectively over trees, in analogy with the 2009 theorem that every nonempty set of regular cardinals is attained by an atomless tree algebra. The two parts constitute one objective, the tower theory of $\operatorname{Treealg}(T)$ beyond regular cardinals; the weak-tower and very-weak-tower (limit ordinal length) formulations are equivalent, and the characterization is sought for every infinite cardinal $\kappa$, the regular case being settled by Monk (2009).

References

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