Partition reals and the consistency of t > add(R)

We show that it is consistent with ZFC that the additivity number add(R) of the ideal of meager sets of the real line is strictly greater than the tower number t of the reals. MSC: 03E35, 54D20.     

Resolutions of Facet Ideals

Abstract

In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.     

Forcing Linearity Numbers for Modules over PID's

Let V be a module over a principal ideal domain. Then V = M N where M is divisible and N has no nonzero divisible submodules. In this paper we determine the forcing linearity number for V when N is a direct sum of cyclic modules. As a consequence, the forcing linearity numbers of several classes of Abelian groups are obtained.     

The weight of a countably compact group whose cardinality has countable cofinality

We show the existence of a p-compact group whose size has countable cofinality in a forcing model. As a corollary, we show that consistently there exists a countably compact group whose size has countable cofinality and its weight is larger than the size.     

Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.     

Multiplicative Mappings of Rings

Let R and F be arbitrary associative rings. A mapping φ of R onto F is called a multiplicative isomorphism if φ is bijective and satisfies φ（xy） = φ（x）φ（y） for all x, y ∈ R. In this short note, we establish a condition on R, in the case where R may not contain any non-zero idempotents, that assures that φ is additive, which generalizes the famous Martindale＇s result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.     

A FORMALISM FOR SOME CLASS OF FORCING NOTIONS

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

Many countable support iterations of proper forcings preserve Souslin trees

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.     

Covering the Baire space by families which are not finitely dominating

It is consistent (relative to ZFC) that each union of many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter , the cofinality of the reduced ultrapower is greater than . The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.     

Forcing closed unbounded subsets of ω2

It is shown that there is no satisfactory first-order characterization of those subsets of ω₂ that have closed unbounded subsets in ω₁,ω₂ and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ⁺ and for partitions of [κ⁺]², when κ is an infinite cardinal.