Almost free groups and Ehrenfeucht-Fra\"ıssé games for successors of singular cardinals

The Ehrenfeucht-Fraisse-game of length omega_1

Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B which we denote by G_{omega_1}(A,B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1 moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of cardinality <= aleph_1. We prove the following results: Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such that the game G_{omega_1}(A,B) is non-determined. Theorem B: If it is consistent that there is a measura…

半环语义中的Ehrenfeucht-Fraïssé博弈

Ehrenfeucht-Fraïssé Games in Semiring Semantics

本文研究半环语义下Ehrenfeucht-Fraïssé博弈的可靠性与完备性，分析其在特定半环上的适用条件，并提出基于同态到布尔半环的新型同态博弈，证明其对有限和无限格半环上的逻辑等价是可靠且完备的。

逻辑生成对抗网络：通过 Ehrenfeucht–Fraïssé 博弈的对抗学习

Logical GANs: Adversarial Learning through Ehrenfeucht Fraisse Games

LOGAN 通过将判别器与生成器分别视为 EF 博弈中的对手和构造者，实现逻辑受限的对抗生成，实验证明其能有效提升属性满足率。

A Framework for Forcing Constructions at Successors of Singular Cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…

EF equivalent not isomorphic models

We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins

Universal graphs at the successor of a singular cardinal

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…

On invariants for omega_1-separable groups

We study the classification of omega_1-separable groups using Ehrenfeucht-Fraisse games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming diamond.

Games and elementary equivalence of $\rm II_1$ factors

We use Ehrenfeucht-Fraïssé games to give a local geometric criterion for elementary equivalence of II$_1$ factors. We obtain as a corollary that two II$_1$ factors are elementarily equivalent if and only their unitary groups are elementarily equivalent as $\mathbb Z_4$-metric spaces.

关于无阿贝尔正规子群的群的描述复杂性

On the Descriptive Complexity of Groups without Abelian Normal Subgroups

本文研究无阿贝尔正规子群的有限群的描述复杂性，提出2-ary WL着色并证明其等价于Hella层级中第二个Ehrenfeucht–Fraïssé双射卵石博弈，进而证明仅需7个卵石和7轮即可识别所有此类群，对应于一阶逻辑中仅用7个变量和7个量词深度的广义2-ary量词公式。

More on regular reduced products

The authors show, by means of a finitary version square^{fin}_{lambda,D} of the combinatorial principle square^{b^*}_{lambda}, the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal lambda, if M_i and N_i are elementarily equivalent models of a language of size <= lambda, then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda^+ on prod_i M_i/D and prod_i N_i/D. If in addition 2^{lambda}= lambda^+ and i<lambda implies |M_i|+|N_i| <= lambda^+ this means that the ultra…

On regular reduced products

Assume <aleph_0,aleph_1>-><lambda,lambda^+>. Assume M is a model of a first order theory T of cardinality at most lambda^+ in a vocabulary L(T) of cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be a set of first order formulas in L(T) and let D be a regular filter on lambda. Then M is Delta-embeddable into the reduced power N^lambda/D, provided that every Delta-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over lambda, M^lambda/D is lambda^{++}-universal. Our second result is as follo…

Games on AF-algebras

We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the m…