A syntactic approach to the MacNeille completion of $\bold\Lambda^{\ast}$, the free monoid over an ordered alphabet $\bold \Lambda$

TL;DR

This paper explores the MacNeille completion of the free monoid over an ordered alphabet, providing a syntactic method to identify closed upper sets and applying this to embedding theorems for valuated oriented graphs.

Contribution

It introduces a syntactic approach to characterize closed upper sets in the MacNeille completion of the free monoid over an ordered alphabet, with applications to graph embeddings.

Findings

Closed upper sets can be generated by binary operations for certain classes of alphabets.

An efficient procedure for testing closedness of upper sets is developed.

Embedding theorems relate closed upper sets to valuated oriented graphs.

Abstract

Let $\Lambda^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $\Lambda$, which is equipped with some (partial) order. This ordering lifts to $\Lambda^{\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $\Lambda^{\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in $\Lambda^*$ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $\Lambda$, one can easily identify the closed lower sets as the finitely generated ones, whereas it is more complicated to determine the closed upper sets. For a fairly large class of ordered sets $\Lambda$ (including complete lattices as well as antichains) one can generate the closure of any upper set of words by means of binary operations ( "syntactic rules") thus obtaining an efficient procedure to test closedness. Closed upper set of words are involved in an embedding theorem for valuated oriented graphs. In fact, generalized paths (so-called "zigzags") are encoded by words over an alphabet $\Lambda$. Then the valuated oriented graphs which are "isometrically" embeddable in a product of zigzags have the characteristic property that the words corresponding to the zigzags between any pair of vertices form a closed upper set in $\Lambda$.