Pro-aperiodic monoids via saturated models

Abstract

We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ω-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.

A Duality Theoretic View on Limits of Finite Structures

“…Equation (10) implies the existence of p, q satisfying P ≥q a ∈ F 1 , P ≥p a / ∈ F 2 and q ≥ p. It follows by (L1) that P ≥p a ∈ F 1 . We conclude that P ≥p a ∈ F 1 \ F 2 , whence F 1 ⊆ F 2 .…”

“…Indeed, it is equal to Typ 0 (T fin ), the space of pseudofinite T -structures. For an application of this, see [10]. Below, we will see an application in finite model theory of the case T = ∅ (in this case we write FO(σ) and Typ(σ) instead of FO(∅) and Typ(∅)).…”

“…Thus, logic fragments are considered modulo the theory of finite words and the Lindenbaum-Tarski algebras are subalgebras of ℘ (A * ) consisting of the appropriate L ϕ 's, cf. [10] for a treatment of first-order logic on words.…”

“…Here, being beyond the scope of this paper, we are ignoring the important role of the monoid structure available on the spaces (in the form of profinite monoids or BiMs, cf [10,5]…”

The linear nature of pseudowords

“…While this paper was being written, Gool and Steinberg developed a different approach on the pseudowords over A, applying Stone duality and model theory to view them as elementary equivalence classes of labeled linear orders [21]. They worked specially with saturated models.…”