Existence of monoids compatible with a family of mappings

TL;DR

This paper explores how to construct monoid structures compatible with specific families of mappings, enabling these mappings to act as left translations, with applications to addition and list concatenation.

Contribution

It introduces a method to derive monoid operations compatible with given mappings, unifying various algebraic structures under a common framework.

Findings

Monoid operations can be systematically constructed for given families of mappings.

The approach applies to addition on natural numbers and integers.

It also extends to list concatenation operations.

Abstract

These notes present an approach to obtaining monoid operations which are compatible with a given family of mappings in the sense that the mappings become left translations in the monoid. This can be applied to various situations such as the addition on the natural numbers and the integers as well as the concatenation of lists.