Hypercodes in deterministic and slender 0L languages

In an infinite sequence of words over a finite alphabet some word must be embedded in a later word. In a D0L sequence such an embedding leads to a decomposition of the language into a finite language and a finite number of DOL languages for which the associated sequences are embedding chains. A language possessing such a decomposition has a bound for the size of the hypercodes (= embedding anti-chains) it can contain. Certain largest hypercodes related to D0L systems and languages are characterized. These characterizations provide methods for showing that languages are not D0L and for showing that pairs of D0L systems are not equivalent. A new class of 0L languages, the slender languages, is introduced. This class contains the finite 0L languages and the deterministic 0L languages. Our key decomposition results are stated for slender languages