zbMATH — the first resource for mathematics

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.
06E05 Structure theory of Boolean algebras
Full Text: DOI
[1] Alaev P.E.: Complexity of Boolean algebras and their Scott rank. Algebra Log. 38(643-666), 768 (1999) · Zbl 0937.03071 · doi:10.1007/BF02671732
[2] Camerlo R., Gao S.: The completeness of the isomorphism relation for countable Boolean algebras. Trans. Amer. Math. Soc. 353, 491–518 (2001) · Zbl 0960.03041 · doi:10.1090/S0002-9947-00-02659-3
[3] Heindorf L.: Alternative characterizations of finitary and well-founded Boolean algebras. Algebra Universalis 29, 109–135 (1992) · Zbl 0753.06013 · doi:10.1007/BF01190759
[4] Kach A.M.: Depth zero Boolean algebras. Trans. Amer. Math. Soc. 362, 4243–4265 (2010) · Zbl 1203.03057 · doi:10.1090/S0002-9947-10-05002-6
[5] Ketonen J.: The structure of countable Boolean algebras. Ann. of Math. 2(108), 41–89 (1978) · Zbl 0418.06006 · doi:10.2307/1970929
[6] Marker D.: Model Theory. An Introduction. Graduate Texts in Mathematics, vol. 217. Springer, New York (2002) · Zbl 1003.03034
[7] Pierce, R.S.: Countable Boolean algebras. In: Handbook of Boolean Algebras, vol. 3, pp. 775-876. North-Holland, Amsterdam (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.