Morphic Primitivity and Alphabet Reductions

Abstract

Abstract. An alphabet reduction is a 1-uniform morphism that maps a word to an image that contains a smaller number of different letters. In the present paper we investigate the effect of alphabet reductions on morphically primitive words, i. e., words that are not a fixed point of a nontrivial morphism. Our first main result answers a question on the existence of unambiguous alphabet reductions for such words, and our second main result establishes whether alphabet reductions can be given that preserve morphic primitivity. In addition to this, we study Billaud's Conjecture -which features a different type of alphabet reduction, but is otherwise closely related to the main subject of our paper -and prove its correctness for a special case.

On Billaud Words and Their Companions

“…Walter [14] identified and proved some of the cases necessary for showing that the conjecture holds for alphabet size of 4. Nevisi and Reidenbach [11] proved that the conjecture is correct for all words (with three or more different letters) if they contain each letter exactly twice.…”