In this paper, we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A*, then the right and left ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from a nonempty word. We show that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, we prove that for any given pseudostandard word s with seed, all sufficiently long left special factors of s are prefixes of it.

A decomposition of a set X of words over a d-letter alphabetA = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets X i , i = 1,...,d, are pairwise disjoint,their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi , where ~denotes the commutative equivalence relation. We introduce some suitable decompositionsthat we call good, admissible, and normal. A normal decomposition is admissible andan admissible decomposition is good. We prove that a set is commutativelyprefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli setswhich have no good decomposition. Moreover, we show that the classical conjecture ofcommutative equivalence of finite maximal codes to prefix ones is equivalent to the statementthat any finite and maximal code has an admissible decomposition.

The characteristic parameters K w and R w of a word w over a finite alphabet are defined as follows: K w is the minimal natural number such that w has no repeated suffix of length K w and R w is the minimal natural number such that w has no right special factor of length R w . In a previous paper, published on this journal, we have studied the distributions of these parameters, as well as the distribution of the maximal length of a repetition, among the words of each length on a given alphabet. In this paper we give the exact values of these distributions in a special case. However, these values give upper bounds to the distributions in the general case. Moreover, we study the most frequent and the average values of the characteristic parameters and of the maximal length of a repetition over the set of all words of length n.

For any finite word w on a finite alphabet, we consider the basic parameters R w and K w of w defined as follows: R w is the minimal natural number for which w has no right special factor of length R w and K w is the minimal natural number for which w has no repeated suffix of length K w . In this paper we study the distributions of these parameters, here called characteristic parameters, among the words of each length on a fixed alphabet.