Abstract
Let h and k be integers greater than 1; we prove that the following statements are equivalent: 1) the direct product of h copies of the additive semigroup of non-negative integers is not k-repetitive; 2) if the direct product of h finitely generated semigroups is k-repetitive, then one of them is finite. Using this and some results of Dekking and Pleasants on infinite words, we prove that certain repetitivity properties are finiteness conditions for finitely generated semigroups.
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References
T. C. Brown, An interesting method in the theory of locally finite semigroups, Pacific J. Math., 36, 285–289(1971).
F.M. Dekking, Strongly non repetitive sequences and progression-free sets, J. Comb. Theory, A, 27, 181–185 (1979).
J. Justin, Propriétés combinatoires de certains semigroupes, C. R. Acad. Paris, A, 269, 1113–1115 (1969).
J. Justin, Semigroupes à générations bornées, dans “Problèmes Mathématiques de la Théorie des Automates”, Séminaire Schützenberger, Lentin, Nivat 69/70, Institut Henri Poincaré, Paris, exposé n. 7, 10 p. (1970).
J. Justin, Sur une Construction de Bruck and Reilly, Semigroup Forum, 3, 148–155 (1971).
J. Justin, Groupes et semigroupes à croissance linéaire, C. R. Acad. Sci. Paris, A, 273, 212–214 (1971).
J. Justin, Semigroupes répétitif, dans “Logique et Automates”, Séminaire I.R.I.A., Institut de Recherche d’Informatique et d’Automatique, Le Chesnay, France, 101–108(1971).
J. Justin, Généralisations du théorème de van der Waerden sur les semigroupes répétitifs, J. Comb. Theory, 12, 357–367 (1972).
J. Justin, Characterization of the repetitive commutative semigroups, J. Algebra, 21, 87–90(1972).
J. Justin, Groupes linéaires répétitifs, C. R. Acad. Sci. Paris, I, 292, 349–350(1981).
J. Justin and G. Pirillo, Two combinatorial properties of partitions of the free semigroup into finitely many parts, Discrete Mathematics, 52, 299–303(1984).
J. Justin and G. Pirillo, On a Natural Extension of Jacob’s Ranks, J. Comb. Theory, 43, 205–218(1986).
J. Justin, G. Pirillo and S. Varricchio, Unavoidable regularities and finiteness conditions for semigroups, Proceedings of the Third Italian Conference “Theoretical Computer Science”, Mantova, 2–4 november 1989, Edited by Bertoni-Bohm-Miglioli, World Scientific (1989).
M Lothaire, Combinatorics on words, Addison-Wesley, 1983.
P. A. B. Pleasants, Non-repetitive sequences, Proc. Cambridge Philos. Soc., 68, 267–274(1970).
G. Pirillo, The van der Waerden Theorem and the Burnside Problem for semigroups, Arch. Math., 53, 1–3(1989).
G. Pirillo, Sur les produits directs de semigroupes répétitifs, submitted.
M. P. Schützenberger, Quelques Problèmes combinatoires de la théorie des automates (J.-F. Perrot Ed.), Cours professé à l’Institut de Programmation, Fac. Sciences de Paris, 1966/67.
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Pirillo, G. Properties of Integers and Finiteness Conditions for Semigroups. Results. Math. 24, 168–173 (1993). https://doi.org/10.1007/BF03322326
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DOI: https://doi.org/10.1007/BF03322326