### On some properties of the syntactic semigroup of a very pure subsemigroup

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A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\{{a}_{1},...,{a}_{d}\}$ is any sequence ${X}_{1},...,{X}_{d},{Y}_{1},...,{Y}_{d}$ of subsets of ${A}^{*}$ such that the sets ${X}_{i}$, $i=1,...,d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, ${X}_{i}\sim {a}_{i}{Y}_{i}$, where $\sim $ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular,...

A decomposition of a set of words over a -letter alphabet is any sequence of subsets of such that the sets , are pairwise disjoint, their union is , and for all , , where ~ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular,...

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...

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.

A code X over the alphabet A is complete if the submonoid X* generated by X meets all two-sided ideals of A*. If one measures the cost of a finite code X over A, with respect to a given information source S, by the quantity gamma(X) = <X> ln |A|, we say that X is completely optimal for S if it does not exist any code X', over an arbitrary alphabet, such that gamma (X') < gamma (X). One can show that for |X| ≤ 5 a completely optimal code has to be complete. However for |X| >...

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

The characteristic parameters and of a word over a finite alphabet are defined as follows: is the minimal natural number such that has no repeated suffix of length and is the minimal natural number such that has no right special factor of length . In a previous paper, published on this journal, we have studied the distributions of these parameters, as well as the distribution of the maximal...

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 is an involutory antimorphism of , then the right and left -palindromic closures of any factor of a -standard word are also factors of some -standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from...

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