Morphically primitive words

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 wellchosen 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.

On the Dual Post Correspondence Problem

“…If α is not morphically imprimitive, we call α morphically primitive. As demonstrated by Reidenbach and Schneider [12], the partition of the set of all patterns into morphically primitive and morphically imprimitive ones is vital in several branches of combinatorics on words and formal language theory, and some of our results in the main part of the present paper shall again be based on this notion.…”

“…Our first criterion is based on so-called ambiguity factorisations, which are a generalisation of imprimitivity factorisations used by Reidenbach and Schneider [12] to characterise the morphically primitive patterns.…”

Restricted Ambiguity of Erasing Morphisms

“…Head [3]). Note that set of succinct patterns is also equivalent to the set of morphically primitive words (as introduced by Reidenbach and Schneider [11]). Regarding the unambiguity of nonerasing morphisms, the classification of patterns into succinct and prolix patterns is vital: According to this result, for any prolix pattern α, every nonerasing morphism is ambiguous.…”

Existence and Nonexistence of Descriptive Patterns

“…In accordance with Reidenbach and Schneider [9], we designate a terminalfree pattern α ∈ X + as morphically imprimitive if there is a pattern β ∈ X * satisfying the following conditions: |β| < |α| and there are morphisms g, h : X * → X * such that g(α) = β and h(β) = α. Otherwise, α is morphically primitive.…”

“…Theorem 3 (Reidenbach, Schneider [9]). A pattern α ∈ X + is morphically primitive if and only if there is no factorisation α = β 0 γ 1 β 1 γ 2 β 2 .…”