aleph_n-Free abelain group with no non-zero homomorphism Z

Discontinuous Homomorphisms of $C(X)$ with $2^{\aleph_0}>\aleph_2$

Assume that $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(ω_1,2)$-morass, $P\in M$ is the poset adding $\aleph_3$ generic reals and $G$ is $P$-generic over $M$. In $M$ we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$-linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $ω$, into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph_0}=\aleph_3$ and, for any infinite compact Hausdorff space $X$, there exists a dis…

Quite free complicated abelian group, PCF and Black Boxes

We like to build Abelian groups (or R-modules) which on the one hand are quite free, say $\aleph_{ω+ 1}$-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism to $Z$ (known classically for $\aleph_1$-free, recently for $\aleph_n$-free). We succeed to prove the existence of even $\aleph_{ω_1 \cdot n}-$free ones. This requires building n-dimensional black boxes, which are quite free. Thus combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardi…

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_α$, then the set has size $\aleph_α$ for any regular $\aleph_α$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposi…

基数阿列夫一下 MV-谱问题的一个解

A solution to the MV-spectrum Problem in size aleph one

本文证明了可数阿贝尔 $\ell$-群主理想格的闭同态提升定理，刻画了至多 $\aleph_1$ 基数的分配 0-格同构条件，从而解决了该基数下的 Mundici MV-谱问题，并将结果推广至森林索引交换图及任意可数全序除环上的向量格。

Some Nasty Reflexive Groups

In "Almost Free Modules, Set-theoretic Methods", Eklof and Mekler raised the question about the existence of dual abelian groups G which are not isomorphic to Z+G. Recall that G is a dual group if G ~ D^* for some group D with D^*=Hom(D,Z). The existence of such groups is not obvious because dual groups are subgroups of cartesian products Z^D and therefore have very many homomorphisms into Z. If p is such a homomorphism arising from a projection of the cartesian product, then D^* ~ ker(p)+Z. In all `classical cases' of groups D of infinite rank it turns out that D^* ~ ker(p). Is this always th…

An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction

We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the know…