# Free monoids and generalized metric spaces

## TL;DR

This paper proves that certain submonoids of final segments over an ordered alphabet are free, and interprets this freeness within the context of generalized metric spaces, revealing structural insights and connections.

## Contribution

It establishes the freeness of submonoids of final segments and their MacNeille completions, and interprets these results in the framework of metric spaces over Heyting algebras.

## Key findings

- The submonoid of non-empty final segments is free.
- The MacNeille completion's non-empty part is also free.
- Final segments correspond to injective envelopes in metric space categories.

## Abstract

Let $A$ be an ordered alphabet, $A^{\ast}$ be the free monoid over $A$ ordered by the Higman ordering, and let $F(A^{\ast})$ be the set of final segments of $A^{\ast}$. With the operation of concatenation, this set is a monoid. We show that the submonoid $F^{\circ}(A^{\ast}):= F(A^{\ast})\setminus \{\emptyset\}$ is free. The MacNeille completion $N(A^{\ast})$ of $A^{\ast}$ is a submonoid of $F(A^{\ast})$. As a corollary, we obtain that the monoid $N^{\circ}(A^{\ast}):=N(A^{\ast})\setminus \{\emptyset\}$ is free. We give an interpretation of the freeness of $F^{\circ}(A^{\ast})$ in the category of metric spaces over the Heyting algebra $V:= F(A^{\ast})$, with the non-expansive mappings as morphisms. Each final segment of $A^{\ast}$ yields the injective envelope $\mathcal S_F$ of a two-element metric space over $V$. The uniqueness of the decomposition of $F$ is due to the uniqueness of the block decomposition of the graph $\mathcal {G}_{F}$ associated to this injective envelope.