Groups elementarily equivalent to a free nilpotent group of finite rank

In this paper we give a complete algebraic description of groups elementarily equivalent to a given free nilpotent group of finite rank

Grothendieck Rings of Theories of Modules

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.Comment: 42 Page

The Tarksi Theorems, Extensions to Group Rings and Logical Rigidity (Logic, Algebraic system, Language and Related Areas in Computer Science)

This is from a talk presented at the Kobe Conference 2022 held in Kobe, Japan.The famous Tarski theorems state that all free groups heve the same elementary theory. In 2019 I gave a talk at the Kobe conference explaining the Tarski theorems and the accompanying language. Subsequently in [FGRS 1, 2, 3] and [FGKRS] the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R was examined. These are relative to an appropriate logical language L₀, L₁, L₂ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS 1]. In [FGRS 1] it was proved that if R[G] is elementarily equivalent to S[H] with respect to L₂, then simultaneously the group G is elementarily equivalent to the group H with respect to Lo, and the ring R is elementarily equivalent to the ring S with respect to L₁. We then let F be a rank 2 free group and Z be the ring of integers. Examining the universal theory of the free group ring Z[F] the hazy conjecture was proved that the universal sentences true in Z[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in Z[F] modified appropriately for group theory. Finally we mention logical group rigidity. A group G is logically rigid if being elementary equivalent to G is equivalent to being isomorphic to G. In this paper we survey all of these findings

Model theory of operator algebras III: Elementary equivalence and II_1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page