Products in a Category with Only One Object

TL;DR

This paper explores decision problems in free Cartesian monoids, introducing a PDA-based model to analyze submonoid properties, with implications for the structure of Thompson-Higman groups.

Contribution

It presents a PDA-based computational model for free Cartesian monoids and applies it to solve key decision problems in specific submonoids, including those relevant to Thompson-Higman groups.

Findings

Decidable membership problems for certain submonoids

Infinite submonoid characterization within the free Cartesian monoid

Application of the model to Thompson-Higman groups

Abstract

We consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us to solve problems such as (1) Given a finite set B of elements and an element F, is F a product of members of B? (2) Is the submonoid generated by the finite set B infinite? for certain fragments of the free Cartesian monoid. These fragments include the submonoid of right invertible elements and so our results apply to the Thompson-Higman groups.