Mobi algebra as an abstraction to the unit interval and its comparison to rings

TL;DR

This paper introduces mobi algebras, an algebraic structure modeling properties of the unit interval relevant to geodesic paths, and explores their relationship with rings and IMM algebras.

Contribution

It defines mobi algebras and establishes their connection with involutive medial monoids and unitary rings, providing a new algebraic framework for geodesic path analysis.

Findings

Every unitary ring with one half corresponds to a mobi algebra with one double.

Mobi algebras capture key properties of the unit interval for geodesic studies.

The paper establishes algebraic relationships between mobi algebras, IMM algebras, and rings.

Abstract

We begin by introducing an algebraic structure with three constants and one ternary operation to which we call mobi algebra. This structure has been designed to capture the most relevant properties of the unit interval that are needed in the study of geodesic paths. Another algebraic structure, called involutive medial monoid (IMM), can be derived from a mobi algebra. We prove several results on the interplay between mobi algebras, IMM algebras and unitary rings. It turns out that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. This paper is the second of a planned series of papers dedicated to the study of geodesic paths from an algebraic point of view.