A characterization of fine words over a finite alphabet

To any infinite word t over a finite alphabet A we can associate two infinite words min(t) and max(t) such that any prefix of min(t) (resp. max(t)) is the lexicographically smallest (resp. greatest) amongst the factors of t of the same length (see Pirillo [14]). In the recent paper [15], Pirillo defined fine words over two letters; specifically, an infinite word t over a 2-letter alphabet {a, b} (a < b) is said to be fine if (min(t), max(t)) = (as, bs) for some infinite word s. Pirillo [15] characterized these words, and remarked that perhaps his characterization can be generalized to an arbitrary finite alphabet; we do just that in this paper. Firstly, we extend the definition of a fine word to more Email address: amy.glen@gmail.com (Amy Glen). than two letters. That is, we say that an infinite word t over A is fine if there exists an infinite word s such that, for any lexicographic order, min(t) = as where a = min(A). Roughly speaking, our main result states that an infinite word t is fine if and only if t is either a strict episturmian word or a strict "skew episturmian word" (i.e., a particular kind of non-recurrent infinite word, all of whose factors are finite episturmian).

Let A denote a finite alphabet. A (finite) word over A is an element of the free monoid A * generated by A, in the sense of concatenation. The identity ε of A * is called the empty word, and the free semigroup, denoted by A + , is defined by A + := A * \ {ε}.

Given w = x 1 x 2 • • • x m ∈ A + with each x i ∈ A, the length of w is |w| = m (note that |ε| = 0). The reversal w of w is given by w = x m x m-1 • • • x 1 , and if w = w, then w is called a palindrome.

An infinite word (or simply sequence) x is a sequence indexed by N with values in A, i.e., x = x 0 x 1 x 2 • • • with each x i ∈ A. The set of all infinite words over A is denoted by A ω , and we define A ∞ := A * ∪ A ω . An ultimately periodic infinite word can be written as uv ω = uvvv • • • , for some u, v ∈ A * , v = ε. If u = ε, then such a word is periodic. An infinite word that is not ultimately periodic is said to be aperiodic.

, and we write w ≺ p z (resp. w ≺ s z). We say that w is a proper factor (resp. prefix, suffix) of

For x ∈ A ω , F (x) denotes the set of all its factors, and F n (x) denotes the set of all factors of x of length n ∈ N, i.e., F n (x) := F (x) ∩ A n . Moreover, the alphabet of x is Alph(x) := F (x) ∩ A, and we denote by Ult(x) the set of all letters occurring infinitely often in x. Any two infinite words x, y ∈ A ω are said to be equivalent if F (x) = F (y), i.e., if x and y have the same set of factors. A factor of an infinite word x is recurrent in x if it occurs infinitely many times in x, and x itself is said to be recurrent if all of its factors are recurrent in it.

Suppose the alphabet A is totally ordered by the relation <. Then we can totally order A * by the lexicographic order <, defined as follows. Given two words u, v ∈ A + , we have u < v if and only if either u is a proper prefix of v or u = xau ′ and v = xbv ′ , for some x, u ′ , v ′ ∈ A * and letters a, b with a < b. This is the usual alphabetic ordering in a dictionary, and we say that u is lexicographically less than v. This notion naturally extends to A ω , as follows.

, where u j , v j ∈ A. We define u < v if there exists an index i ≥ 0 such that u j = v j for all j = 0, . . . , i -1 and u i < v i . Naturally, ≤ will mean < or =.

Let w ∈ A ∞ and let k be a positive integer. We denote by min(w|k) (resp. max(w|k)) the lexicographically smallest (resp. greatest) factor of w of length k for the given order (where |w| ≥ k for w finite). If w is infinite, then it is clear that min(w|k) and max(w|k) are prefixes of the respective words min(w|k + 1) and max(w|k + 1). So we can define, by taking limits, the following two infinite words (see [14]) min(w) = lim k→∞ min(w|k) and max(w) = lim k→∞ max(w|k).

A morphism on A is a map ψ : A * → A * such that ψ(uv) = ψ(u)ψ(v) for all u, v ∈ A * . It is uniquely determined by its image on the alphabet A. All morphisms considered in this paper will be non-erasing: the image of any non-empty word is never empty. Hence the action of a morphism ψ on A * naturally extends to infinite words; that is, if

The free monoid A * can be naturally embedded within a free group. We denote by F (A) the free group over A that properly contains A, and is obtained from A by adjoining the inverse a -1 of each letter a ∈ A. More precisely, we construct a new alphabet A ± that consists of all letters a of A and their ‘inverses’ a -1 , i.e., A ± = {a, a -1 | a ∈ A}. If one defines on the free monoid (A ± ) * the involution (a -1 ) -1 = a for each a ∈ (A ± ) * , then necessarily, we have (uv) -1 = v -1 u -1 for all u, v ∈ (A ± ) * . The free group F (A) over A is the quotient of (A ± ) * under the relation: aa -1 = a -1 a = ε for all a ∈ A. In what follows, we use the notation p -1 w and ws -1 to indicate the removal of a prefix p (resp. suffix s) from a finit

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