Unique decipherability in the additive monoid of sets of numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 45 (2011) no. 2, pp. 225-234.

Sets of integers form a monoid, where the product of two sets A and B is defined as the set containing a+b for all aA and bB. We give a characterization of when a family of finite sets is a code in this monoid, that is when the sets do not satisfy any nontrivial relation. We also extend this result for some infinite sets, including all infinite rational sets.

DOI : https://doi.org/10.1051/ita/2011018
Classification : 68R05,  68Q45
Mots clés : unique decipherability, rational set, sumset
@article{ITA_2011__45_2_225_0,
     author = {Saarela, Aleksi},
     title = {Unique decipherability in the additive monoid of sets of numbers},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {225--234},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {2},
     year = {2011},
     doi = {10.1051/ita/2011018},
     zbl = {1218.68108},
     mrnumber = {2811655},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2011018/}
}
Saarela, Aleksi. Unique decipherability in the additive monoid of sets of numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 45 (2011) no. 2, pp. 225-234. doi : 10.1051/ita/2011018. http://www.numdam.org/articles/10.1051/ita/2011018/

[1] J. Berstel and D. Perrin, Theory of Codes. Academic Press (1985). | MR 797069 | Zbl 0587.68066

[2] A. Brauer, On a problem of partitions. Amer. J. Math. 64 (1942) 299-312. | MR 6196 | Zbl 0061.06801

[3] Ch. Choffrut and J. Karhumäki, Unique decipherability in the monoid of languages: an application of rational relations, in Proceedings of the Fourth International Computer Science Symposium in Russia (2009) 71-79. | Zbl 1248.94044

[4] R. Gilmer, Commutative Semigroup Rings. University of Chicago Press (1984). | MR 741678 | Zbl 0566.20050

[5] J.-Y. Kao, J. Shallit and Z. Xu, The frobenius problem in a free monoid, in Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science (2008) 421-432. | MR 2873754 | Zbl 1259.68166

[6] J. Karhumäki and L.P. Lisovik, The equivalence problem of finite substitutions on ab * c, with applications. Int. J. Found. Comput. Sci. 14 (2003) 699-710. | MR 2010592 | Zbl 1101.68660

[7] M. Kunc, The power of commuting with finite sets of words. Theor. Comput. Syst. 40 (2007) 521-551. | MR 2305376 | Zbl 1121.68065

[8] D. Perrin, Codes conjugués. Inform. Control. 20 (1972) 222-231. | MR 345711 | Zbl 0254.94015

[9] J.L. Ramírez Alfonsín, The Diophantine Frobenius Problem. Oxford University Press (2005). | MR 2260521 | Zbl 1134.11012