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Downward closure of depth in countable Boolean algebras. (English) Zbl 1260.06016
The depth of a countable Boolean algebra (BA) is defined as follows. If $$A$$ is superatomic, one considers the sequence of ideals on $$A$$: $$I_0=\{0\}$$, $$I_{\alpha+1}$$ is the ideal on $$A$$ consisting of $$a\in A$$ such that $$a/I_\alpha$$ is a finite sum of atoms, and $$I_\gamma=\bigcup_{\beta<\gamma}I_\beta$$ for $$\gamma$$ limit. The rank of $$A$$ is the least ordinal $$\alpha$$ such that $$I_{\alpha+1}=A$$; this is a successor ordinal. A measure on a countable BA $$A$$ is a mapping from $$A$$ to some monoid $$M$$ satisfying some simple conditions. For any countable BA $$A$$ and $$a\in A$$ we define $$\mu_A(a)=\sup\{\text{rank}(b):b\leq a$$ and $$A|b$$ is superatomic$$\}$$. Then the canonical measure on $$A$$ is defined by $$\sigma_A(a)=-1$$ if $$A|a$$ is superatomic, and $$\sigma_A(a)=\min\{\mu_A(b):b\leq a$$ and $$A|(a\cdot-b$$ is superatomic$$\}$$ for $$A|a$$ not superatomic. This is a measure with monoid $$\{-1\}\cup\omega_1$$. Given a measure $$\mu:A\rightarrow M$$ one defines the derived measure $$D\mu$$ by letting $$(D\mu)(a)$$ be the set of all finite sequences $$\langle\mu(a_0),\dots,\mu(a_{m-1})\rangle$$ with $$a$$ the disjoint sum of all $$a_i$$ (some of which may be 0). This operation $$D$$ can be repeated, even transfinitely many times. The measure $$D$$ is stable if $$\mu(a)=\mu(b)$$ implies $$(D\mu)(a)=(D\mu)(b)$$ for all $$a,b\in A$$. The depth of $$A$$ is the least $$\alpha$$ such that $$D^\alpha$$ is stable.
This paper studies this notion of depth. Some results are: There is a countable BA whose depths of relative algebras are precisely the sets $$\{0,\dots,\alpha,\alpha+2\}$$ for each ordinal $$\alpha$$; or $$\{0,1,\omega^\alpha\}$$, or $$\{0,2,\omega^\alpha\}$$. Every countable BA of depth at least one has a depth-zero relative algebra, and either a depth-one relative algebra or a depth-2 relative algebra.
##### MSC:
 600000 Structure theory of Boolean algebras
##### Keywords:
countable Boolean algebras; depth
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