Monoids and Maximal Codes

TL;DR

This paper extends the concept of maximality to non-uniquely decipherable codes by introducing full monoids within a partial order of submonoids, providing a new framework for understanding maximal codes.

Contribution

It introduces a partial order on monoids, defines full monoids as maximal elements, and extends the notion of maximality to non-UD codes with a characteristic property.

Findings

Existence of maximal elements called full monoids in the poset of submonoids.

A set of generators of a full monoid is a maximal code.

Extension of the maximality concept from UD to non-UD codes.

Abstract

In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code.