Abstract. Motivated by the research on pattern languages, we study a fundamental combinatorial question on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism ρ with ρ = σ and ρ(α) = σ(α). Our main result shows that a rich and natural class of strings is provided with unambiguous morphic images.

This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ (α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601-628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.

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 his item ws sumitted to voughorough niversity9s snstitutionl epository y theGn uthorF Citation: heD tFhFD ishixfegrD hF nd grxishiD tFgFD PHIQF yn the dul post orrespondene prolemF sxX f¡ elD wFF nd grtonD yF @edsAF hevelopments in vnguge heoryX IUth snterntionl gonfereneD hv PHIQD wrneElEll¡ eeD prneD tune IVEPID PHIQD roeedingsF veture xotes in gomE puter iene @inluding suseries veture xotes in ertifiil sntelligene nd veture xotes in fioinformtisAY UWHUD ppFITUEIUVF Abstract. The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ, τ , one of which needs to be non-periodic, such that σ(α) = τ (α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only.Little is known about the Dual PCP for words α over larger than binary alphabets. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist.

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