Geodesic
Multipotent sets in homogeneous commutative monoids
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 27-36.

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In this paper, we introduce the concept of $k$-potent sets in monoids, $k\in\mathbb{N}$, establish their simplest properties, and indicate a class of homogeneous monoids with a set of generating elements. We find simple necessary conditions of the $k$-potency of a fixed set in such a monoid. For commutative monoids, we establish an isormorphism between them and the monoid $\mathbb{N}_+^{\mathfrak{J}}$ with the corresponding label set $\mathfrak{J}$. For commutative homogeneous monoids with sets of generators, we prove necessary and sufficient conditions for the $k$-potency of their subsets. Finally, we apply this result to the binary Goldbach problem in analytic number theory.
Keywords: commutativity, monoid, homogeneity, prime number, cycle.
Mots-clés : multipotent set 
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Yu. P. Virchenko. Multipotent sets in homogeneous commutative monoids. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 27-36. http://geodesic.mathdoc.fr/item/INTO_2022_204_a2/

[1] Burbaki N., Elementy matematiki. Algebra. Chast 1. Algebraicheskie struktury. Lineinaya i polilineinaya algebra, Fizmatlit, M., 1962

[2] Vinogradov I. M., Osnovy teorii chisel, Gostekhizdat, M.-L., 1952 | MR

[3] Lyapin E. S., Polugruppy, Fizmatlit, M., 1960 | MR

[4] Prakhar K. P., Raspredelenie prostykh chisel, Mir, M., 1967

[5] Chandrasekkharan K., Vvedenie v analiticheskuyu teoriyu chisel, Mir, M., 1974

[6] Shevrin L. N., “Polugruppy”, Spravochnaya matematicheskaya biblioteka. Obschaya algebra, v. 2, Nauka, M., 1991, 11–191

[7] Goldbach C., “Lettre XLIII”, Correspondance mathematique et physique de quelques celebres geometres du XVIII-eme siecle, v. 1, St. Petersbourg, 1743, 125–129

[8] Waring E., Meditationes Algebraicae, Cambridge, 1770 | MR