Multifraction reduction III: The case of interval monoids

TL;DR

This paper studies gcd-monoids, focusing on interval monoids from finite posets, revealing complex behaviors in multifraction reduction such as semi-convergence without full convergence and non-embeddability into their enveloping groups.

Contribution

It introduces specific gcd-monoids with novel properties in multifraction reduction, highlighting behaviors not previously documented in the literature.

Findings

Existence of gcd-monoids with semi-convergence without full convergence

Construction of gcd-monoids with semi-convergence limited to certain levels

Identification of gcd-monoids that cannot embed into their enveloping groups

Abstract

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).