Morphically primitive words

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 @S, we call a word w@?@S^*morphically imprimitive provided that there are a shorter word v and morphisms h,h^':@S^*->@S^* 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.