Arithmetic and $k$-maximality of the cyclic free magma

TL;DR

This paper investigates the structure of free magmas, introduces a ringoid-like structure with an additional operation, and studies the lattice of submagmas using concepts of arithmetic and $k$-maximality.

Contribution

It introduces a new ringoid-like structure on cyclic free magmas and analyzes the lattice of submagmas through arithmetic and $k$-maximality concepts.

Findings

Defined a ringoid-like structure with a second distributive operation.

Characterized $k$-maximal submagmas within free magmas.

Explored the lattice structure of submagmas in free magmas.

Abstract

We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is $k$-maximal when there are only $k-1$ submagmas between it and the free magma itself. These two tools, arithmetic and maximality, allow us to study the lattice of the submagmas of a free magma.