Lyndon word

A Lyndonwort is a formal word that is lexicographically smaller than any rotation of its letters. Each word can be uniquely decomposed into a lexicographically decreasing sequence of Lyndonwörtern.

Formal definition

A word is a Lyndonwort if and only if for each decomposition with non-empty words and that

Examples

• A single letter is always a Lyndonwort, because it can not be decomposed into two non-empty words, and thus the condition is empty.
• Lyndonwort is not because they are subject to and that.
• Is a Lyndonwort as with and the only decomposition into non-empty words and is valid.

Shirshov decomposition

Each Lyndonwort, which consists of more than just a letter, can be decomposed into two Lyndonwörter and with and. The decomposition shortest- called Shirshov decomposition.

Conversely, also, that for all Lyndonwörter and applies that a Lyndonwort is.

Other examples

• The Shirshovzerlegung of with and.
• Since Lyndonwörter are, are also and Lyndonwörter.
• Also, a Lyndonwort. It can be decomposed into both the Lyndonwörter and as well as in the Lyndonwörter and. Since shorter than is the Shirshovzerlegung of.
• Each Lyndonwort has the structure wherein Lyndonwörter is. In this way, it is easily seen that a Lyndonwort is.