On finitary properties for fiber products of free semigroups and monoids

TL;DR

This paper investigates conditions under which fiber products of free semigroups and monoids are finitely generated or presented, providing criteria based on the properties of fiber quotients and constructing automata for indecomposable elements.

Contribution

It establishes necessary and sufficient conditions for finite generation and presentation of fiber products of free semigroups and monoids, including automata construction for indecomposable elements.

Findings

Fiber products of free monoids are finitely generated if and only if the fiber quotient is finite.

All fiber products of free monoids with finite fiber quotients are finitely presented.

Fiber products of free semigroups over finite quotients are never finitely generated.

Abstract

We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $\mathcal{J}$-trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.