Homogeneity in the free group

Perin, Chloé; Sklinos, Rizos

doi:10.1215/00127094-1813068

Cited by 32 publications

(44 citation statements)

References 27 publications

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“…For example every two free groups F m and F n are locally isomorphic, but not isomorphic provided m = n. As will see F m and F n are not also isotypic. This follows from the result of C. Perin and R. Sklinos [10] on logical homogenity (see Section 2.4) of a free group.…”

Section: Some Results and Problemsmentioning

confidence: 65%

Algebraic logic and logical geometry in arbitrary varieties of algebras

Plotkin¹

2014

Group Theory, Combinatorics, and Computing

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Section: Some Results and Problemsmentioning

confidence: 65%

Algebraic logic and logical geometry in arbitrary varieties of algebras

Plotkin¹

2014

Group Theory, Combinatorics, and Computing

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“…We conclude this section by giving another application of Corollary 3.13. The following result is easily deduced from Proposition 5.9 and Proposition 6.2 of [PS09].…”

Section: Definition 42mentioning

confidence: 79%

“…The following result is Theorem 7.1 of [PS09]. Before stating it we recall that the connectedness of T f g implies that p 0 is stationary (as any non forking extension of p 0 is also a generic type).…”

Section: Hyperbolic Floors and Towersmentioning

confidence: 96%

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Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups

Louder¹,

Perin²,

Sklinos³

2013

Notre Dame J. Formal Logic

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We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type p 0 , but that there is a finitely generated model which omits p (2) 0 . We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.To Anand Pillay on the occasion of his 60th birthday.

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“…The same result is true for A Γ without the center because the standard generating set is definable. Notice that non-abelian free groups are also homogeneous [9], [7]. Definition 2.…”

Section: Definability Of a Submonoidmentioning

confidence: 99%

Bi-interpretability of some monoids with the arithmetic and applications

Kharlampovich

López

2019

Semigroup Forum

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We will prove bi-interpretability of the arithmetic N = N, +, ·, 0, 1 and the weak second order theory of N with the free monoid MX of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet X is definable in MX . Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence φ such that every finitely generated monoid satisfying φ is isomorphic to MX . The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with N to any boolean combination of formulas from Πn or Σn. * Hunter College, CUNY † Graduate Center, CUNYIt follows from Quine's paper [10], Section 4, that for a free monoid of rank n ≥ 2 and generating set X = {x 1 , . . . , x n }, the arithmetic N, +, ·, ↑, 0, 1 is bi-interpretable in M X with parameters X, where x ↑ y means x y . Since the predicate z = x y is computable and therefore definable in terms of addition and multiplication (see, for example, [5]) it can be removed from the signature. Quine was working with the structure of concatenation with parameters, C = C, ⌢ , where C is the set of all finite strings in a finite alphabet and ⌢ is the concatenation operation. This structure C and the free semigroup are equivalent structures. Quine did not state the monoid version of his results but the presence of the identity doesn't make a difference and his results are also valid for M X .We will show that N and the weak second order theory of N are bi-interpretable with M X (with parameters X) using the technique that allows also to prove this for a non-trivial free partially commutative monoid A Γ with trivial center. Here we say that the weak second order theory of a structure B is interpretable in A if the first-order structure S(B, N) (see Section 2.1 for precise definition) which has the same expressive power, is interpretable in A.This bi-interpretability has interesting applications. For example, Theorem 4 states that for any k ∈ N, there is a formula ψ(y, y 1 , . . . , y k , X) such that ψ(g, g 1 , . . . , g k , X) holds in M X if and only if g belongs to the submonoid generated by g 1 , . . . , g k . In other words, finitely generated submonoids of M X are definable. In contrast to this, it was proved in [4] and later in [8] that proper subgroups of a free group are not definable (except cyclic subgroups when the language contains constants). This was a solution of an old problem posed by Malcev. Primitive elements, and, therefore, free bases are not definable in a free group of rank greater than 2 [4], but they are easily definable in the free monoid. This implies that the free monoid is homogeneous (two tuples realize the same types if and only if they are automorphically equivalent). We wil...

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Homogeneity in the free group

Cited by 32 publications

References 27 publications

Algebraic logic and logical geometry in arbitrary varieties of algebras

Algebraic logic and logical geometry in arbitrary varieties of algebras

Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups

Bi-interpretability of some monoids with the arithmetic and applications

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