Morphically Primitive Words (extended Abstract) *

In the present paper, we introduce an alternative notion of the primitivity of words, that – unlike the standard understanding of this term – is not based on the power (and, hence, the concatenation) of words, but on morphisms. For any alphabet Σ, we call a word w ∈ Σ∗ morphically imprimitive provided that there are a shorter word v and morphisms h, h′ : Σ∗ → Σ∗ satisfying h(v) = w and h′(w) = v, and we say that w is morphically primitive otherwise. We explain why this is a well-chosen terminology, we demonstrate that morphic (im-)primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms, and we study a number of fundamental properties of the concepts under consideration.