Bi-interpretability of some monoids with the arithmetic and applications

Abstract

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...

Rich groups, weak second order logic, and applications

“…Quine's results on free mopnoids are generalized in [23] for arbitrary partially commutative monoids with trivil center. Namely, the following result holds.…”

“…Theorem 21. [23] If a free partially commutative monoid A Γ has trivial center, then N and A Γ are bi-interpretable with the generating set V (the set of vertices of Γ) as parameters. If A Γ = M X is a free monoid with basis X, |X| > 1, then this bi-interpretation is uniform in X and M X is absolutely rich.…”

“…[23] The arithmetic N is interpretable (with parameters) in the following classes of monoids:a) Baumslag-Solitar monoids a, b | ab k = b m a with k, m > 2 (we do not need parameters for them); b) Non-commutative free partially commutative monoids; c) One-relator monoids G = a, b, C|x = y , where C is a non-empty alphabet, some letter of C appears in y and neither x nor y end with a. If G is a monoid from Theorem 23, then the first-order theory T h(G) is undecidable.…”