On a special class of primitive words

Abstract

a b s t r a c tWhen representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its Watson-Crick complement θ (u), where θ denotes the Watson-Crick complementarity function. Thus, an expression which involves only a word u and its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called pseudo-primitive words relative to θ or simply θ-primitive words, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique θ-primitive root of a given word, and we give some constraints forcing two distinct words to share their θ-primitive root. Also, we present an extension of the well-known Fine and Wilf theorem, for which we give an optimal bound.

Properties of Pseudo-Primitive Words and Their Applications

“…The notion of primitive word was generalized into that of pseudo-primitive word by Czeizler, Kari, and Seki [4]. For an antimorphic involution θ, a non-empty word w ∈ Σ + is said to be pseudo-primitive with respect to θ, or simply θ-primitive, if w ∈ {v, θ(v)} n implies n = 1 for any word v ∈ Σ + .…”

“…For an antimorphic involution θ, a non-empty word w ∈ Σ + is said to be pseudo-primitive with respect to θ, or simply θ-primitive, if w ∈ {v, θ(v)} n implies n = 1 for any word v ∈ Σ + . In [4] it was proved that for any non-empty word w ∈ Σ + , there exists a unique θ-primitive word t satisfying w ∈ t{t, θ(t)} * . Such a word t is called the θ-primitive root of w. The next lemma describes a property of the θ-primitive root of a θ-palindrome of even length.…”

“…It states that for two words u, v ∈ Σ + , if a power of u and a power of v share a prefix of length at least |u| + |v| − gcd(|u|, |v|), then ρ(u) = ρ(v), where gcd(·, ·) denotes the greatest common divisor of two arguments (for its proof, see, e.g., [2]). This theorem has been generalized in [4], by taking into account the equivalence between a word and its image under θ, in the following two forms.…”

“…In a way, we can say that these theorems describe relations causing a weak defect effect because they all imply that u, v ∈ {t, θ(t)} + for some θ-primitive word t ∈ Σ + , which is strictly weaker than the usual defect effect ρ(u) = ρ(v) [4]. Various relations causing such a weak defect effect were presented in [4].…”

“…Note that for words u, v ∈ Σ * , it was proved in [4] that the system uv = θ(uv) and vu = θ(vu) causes a weak defect effect: u, v ∈ {t, θ(t)} * for some t ∈ Σ + . Thus for words x, y, z satisfying xy = zx, if both y and z are θ-palindromes, then the representation of solutions of xy = zx implies tr = θ(tr) and rt = θ(rt).…”