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

Abstract

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.