Cantor-Bendixson Rank Research Articles

Finite Length and Pure-Injective Modules over a Ring of Differential Operators

Let k be an algebraically closed field of characteristic zero, On=k[[x1,…,xn]] the ring of formal power series over k, and Dn the ring of differential operators over On. Suppose that ρ is a prime ideal of On of height n−1; i.e., A=On/ρ is a curve. We prove that every indecomposable finite length module over Dn with support on ρ is uniserial with isomorphic or alternating composition factors. For the ring D(A) of differential operators over A we also classify indecomposable pure-injective modules and show that the Cantor–Bendixson rank of the Ziegler spectrum over D(A) is equal to 2.

Cantor-Bendixson Rank of the Ziegler Spectrum Over a Commutative Valuation Domain

AbstractWe calculate the Cantor-Bendixson rank of the Ziegler spectrum over a commutative valuation domain R proving that it is equal to the double Krull dimension of R.

Effective presentability of Boolean algebras of Cantor-Bendixson rank 1

AbstractWe show that there is a computable Boolean algebra and a computably enumerable ideal I of such that the quotient algebra /I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.

Rank, join, and Cantor singletons

A Cantor singleton is the unique nonrecursive member of some \(\Pi^0_1\) class. In this paper we investigate the relationships between the following three notions: Cantor singletons, Cantor-Bendixson rank, and recursive join. Among other results, we show that the rank of \(A\oplus B\) is at most the natural sum of the ranks of \(A\) and \(B\), and that, if \(B\) has the same rank as \(A\o plus B\), then \(A\) is recursive in \(B\).

Near-discreteness of modules and spaces as measured by Gabriel and Cantor

The Gabriel dimension analysis of a module category and the Cantor-Bendixson rank analysis of a topological space have a common generalization via a lifting to certain lattices. In many instances the Gabriel derivative and the Cantor-Bendixson derivative coincide.

The Gabriel Dimension and Cantor-Bendixson Rank of a Ring

The Gabriel Dimension and Cantor-Bendixson Rank of a Ring

A note on finitely generated models

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.