In this paper, we give a characterization of injective semimodules over additively idempotent semirings. Consequently, we provide a complete description of injective semimodules over the semifield of tropical integers, and give an explicit construction of the injective hulls of semimodules over chain division semirings. We also give a criterion for self-injective MV-semirings with an atomic Boolean center, and describe the structure of (finitely generated) injective semimodules over finite MV-semirings, as well as we show that every complete MV-semiring with an atomic Boolean center is an exact semiring which is defined by a Hahn-Banach-type separation property on semimodules arising in the tropical case from the phenomenon of tropical matrix duality. Moreover, we show that complete Boolean algebras are precisely the MV-semirings in which every principal ideal is injective.

On injectivity of semimodules over additively idempotent division semirings and chain MV-semirings

Di Nola, A.;Lenzi, G.;VANNUCCI, SARA
2019-01-01

Abstract

In this paper, we give a characterization of injective semimodules over additively idempotent semirings. Consequently, we provide a complete description of injective semimodules over the semifield of tropical integers, and give an explicit construction of the injective hulls of semimodules over chain division semirings. We also give a criterion for self-injective MV-semirings with an atomic Boolean center, and describe the structure of (finitely generated) injective semimodules over finite MV-semirings, as well as we show that every complete MV-semiring with an atomic Boolean center is an exact semiring which is defined by a Hahn-Banach-type separation property on semimodules arising in the tropical case from the phenomenon of tropical matrix duality. Moreover, we show that complete Boolean algebras are precisely the MV-semirings in which every principal ideal is injective.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4727969
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