A Diamond Structure in the Transducer Hierarchy

TL;DR

This paper proves the existence of a diamond structure within the transducer hierarchy, advancing understanding of word transformation classes and introducing a new stream function for further research.

Contribution

It constructs a diamond structure in the transducer hierarchy and introduces a novel function on streams to facilitate future studies.

Findings

Confirmed the existence of a diamond structure in the transducer hierarchy

Developed a new function on streams for analyzing transducer degrees

Provided a construction method for the diamond structure

Abstract

We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite state transducer, which form an order based on which words can be transformed into other words. We provide a construction which proves the existence of a diamond structure, while also introducing a new function on streams which may be useful for proving more results about the transducer hierarchy.