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Elementary embeddings in torsion-free hyperbolic groups

Chloé Perin (2011)

Annales scientifiques de l'École Normale Supérieure

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if H embeds elementarily in a torsion free hyperbolic group Γ , we show that the group Γ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of  H with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the...

Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields

Vincent Astier (2013)

Fundamenta Mathematicae

We prove that the boolean algebras of sets definable in elementarily equivalent o-minimal expansions of real closed fields are back-and-forth equivalent, and in particular elementarily equivalent, in the language of boolean algebras with new predicates indicating the dimension, Euler characteristic and open sets. We also show that the boolean algebra of semilinear subsets of [0,1]ⁿ definable in an o-minimal expansion of a real closed field is back-and-forth equivalent to the boolean algebra of definable...

Embedding sums of cancellative modes into semimodules

Anna B. Romanowska, Anna Zamojska-Dzienio (2005)

Czechoslovak Mathematical Journal

A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice.

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