Free monoids and generalized metric spaces

Let A be an ordered alphabet, A∗ be the free monoid over A ordered by the Higman ordering, and let F(A∗) be the set of final segments of A∗. With the operation of concatenation, this set is a monoid. We show that the submonoid F∘(A∗)≔F(A∗)∖{0̸} is free. The MacNeille completion N(A∗) of A∗ is a submonoid of F(A∗). As a corollary, we obtain that the monoid N∘(A∗)≔N(A∗)∖{0̸} is free. We give an interpretation of the freeness of F∘(A∗) in the category of metric spaces over the Heyting algebra V≔F(A∗), with the non-expansive mappings as morphisms. Each final segment F of A∗ yields the injective envelope SF 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 GF associated to this injective envelope.