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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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/

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