# Some decompositions of Bernoulli sets and codes

• Volume: 39, Issue: 1, page 161-174
• ISSN: 0988-3754

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## Abstract

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A decomposition of a set X of words over a d-letter alphabet A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets Xi, 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 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, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.

## How to cite

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de Luca, Aldo. "Some decompositions of Bernoulli sets and codes." RAIRO - Theoretical Informatics and Applications 39.1 (2010): 161-174. <http://eudml.org/doc/92753>.

@article{deLuca2010,
abstract = { A decomposition of a set X of words over a d-letter alphabet A = \{a1,...,ad\} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets Xi, 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 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, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition. },
author = {de Luca, Aldo},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Bernoulli sets; codes; decompositions; commutative equivalence.; Bernoulli distributions; prefix set},
language = {eng},
month = {3},
number = {1},
pages = {161-174},
publisher = {EDP Sciences},
title = {Some decompositions of Bernoulli sets and codes},
url = {http://eudml.org/doc/92753},
volume = {39},
year = {2010},
}

TY - JOUR
AU - de Luca, Aldo
TI - Some decompositions of Bernoulli sets and codes
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 1
SP - 161
EP - 174
AB - A decomposition of a set X of words over a d-letter alphabet A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets Xi, 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 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, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.
LA - eng
KW - Bernoulli sets; codes; decompositions; commutative equivalence.; Bernoulli distributions; prefix set
UR - http://eudml.org/doc/92753
ER -

## References

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5. G. Hansel, Baïonnettes et cardinaux. Discrete Math.39 (1982) 331–335.  Zbl0486.20032
6. D. Perrin and M.P. Schützenberger, Un problème élémentaire de la théorie de l'Information, in Theorie de l'Information, Colloq. Internat. du CNRS No. 276, Cachan (1977) 249–260.
7. D. Perrin and M.P. Schützenberger, A conjecture on sets of differences on integer pairs. J. Combin. Theory Ser. B30 (1981) 91–93.  Zbl0468.05033
8. J.E. Pin and I. Simon, A note on the triangle conjecture. J. Comb. Theory Ser. A32 (1982) 106–109.  Zbl0472.05010
9. P. Shor, A counterexample to the triangle conjecture. J. Comb. Theory Ser. A38 (1985) 110–112.  Zbl0558.20032

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