Mathematical logic, algebra and number theory

Chainable properties of semigroups of nonnegative matrices

Yu. A. Alpin^a, A. E. Guterman^b, E. R. Shafeev<sup>cd

a</sup> Gogol st. 9, ap. 1, 420015 Kazan, Russia
^b Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
^c Moscow Center of Fundamental and Applied Mathematics, 119991 Moscow, Russia
^d Department of Mathematics and Mechanics, Moscow State University, 119991 Moscow, Russia

Abstract: The theorem by Protasov and Voynov on the combinatorial structure of semigroups of nonnegative matrices extends a well-known result of Frobenius on the canonical form of an irreducible nonnegative matrix. We generalize the Protasov — Voynov theorem to not necessarily irreducible semigroups of matrices. For this purpose, an extensions of the concepts of imprimitivity index and canonical partition are introduced which are based on the chain properties of nonnegative matrices.

Keywords: nonnegative matrices, chainable matrices, chainable index.

UDC: 512.643

MSC: 15B48

Language: English

DOI: 10.33048/semi.2024.21.054