Related papers: Almost free algebras
We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted…
It is shown that the big free group (the set of countably-long words over a countable alphabet) is almost free, in the sense that any function from the alphabet to a compact topological group factors through a homomorphism. This statement…
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite…
For every infinite cardinal $\kappa$ with $\kappa^+=2^\kappa$ we construct a group $G$ of cardinality $|G|=\kappa^+$ such that (i) $G$ is $36$-Shelah, which means that $A^{36}=G$ for any subset $A\subseteq G$ of cardinality $|A|=|G|$; (ii)…
We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…
The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $\kappa$ is singular strong limit, then $2^{\kappa}=\kappa^+$. We prove that given a singular cardinal…
Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext_R(G,G)=0 holds. For simplicity we will call such modules splitters. Our investigation continues math.LO/9910159. In math.LO/9910159, we…
We denote by Conc(A) the semilattice of compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc(V) the class of all semilattices isomorphic to Conc(A) for some A in V. Given varieties V1 and V2 varieties of…
For each cardinal $\kappa$, each natural number $n$ and each simplicial complex $K$ we construct a space $\nu^n_\kappa(K)$ and a map $\pi \colon \nu^n_\kappa(K) \to K$ such that the following conditions are satisfied. 1. $\nu^n_\kappa(K)$…
Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that $\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at most finitely many…
We study the existence and cardinality of universal families for classes of rayless graphs. It is known, by a result of Diestel, Halin, and Vogler, that the class of countable rayless graphs does not admit a countable universal family,…
Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing $\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a…
For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest…
We define a generic Vop\v{e}nka cardinal to be an inaccessible cardinal $\kappa$ such that for every first-order language $\mathcal{L}$ of cardinality less than $\kappa$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if…
Given two infinite cardinals $\kappa$ and $\lambda$, we introduce and study the notion of a $\kappa$-barely independent family over $\lambda.$ We provide some conditions under which these types of families exist. In particular, we relate…
Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…
A family $A_\alpha$ of differential operators depending on a real parameter $\alpha\ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum…
Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$…
In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic…
We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are…