Projective systemic modules

Jun, Jaiung; Mincheva, Kalina; Rowen, Louis

doi:10.1016/j.jpaa.2019.106243

Cited by 8 publications

(4 citation statements)

References 33 publications

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“…One can easily see that with Definition 3.19, any free module is projective. We also note that in [JMR19] and [JMR20], more general versions of projective modules are introduced in a framework of systems which include modules over hyperfields.…”

Section: Free Modules and Projective Modules Over Semiringsmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.

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Section: Free Modules and Projective Modules Over Semiringsmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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“…See [35] for a relatively brief introduction of systems; more details are given in [25], [27], and [34]. Throughout the paper, we let N be the additive monoid of nonnegative integers.…”

Section: Basic Notionsmentioning

confidence: 99%

“…Meanwhile an extensive literature developed around hyperstructures [5,7,16,26,38] and fuzzy rings [10,11]. These constructions were unified by Lorscheid in "blueprints" [30,31], and carried further into a "systemic" algebraic approach taken by [34], and developed in [2,13,25,27] in order to unify classical algebra with the algebraic theories of supertropical algebra, symmetrized semirings, hyperfields, and fuzzy rings, with some success especially in obtaining theorems about matrices, polynomials, and linear algebra.…”

Section: Introductionmentioning

confidence: 99%

$\mathcal{T}$-semiring pairs

Jun¹,

Atanasov²,

Rowen³

2022

Preprint

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We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context. 4.1. Fractions 10 4.2. Integral extensions 10 4.3. Hilbert Nullstellensatz 12 5. Growth in semialgebras 13 5.1. Growth in a pair 13 References 14

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“…Semirings in this case seem to provide a reasonable algebraic structure on which homological algebra in characteristic one can be built as it was shown in [CC19] by Connes and Consani. Also, for an approach which simultaneously deals with homological algebra for semirings and hyperfields, we refer the reader to [JMR19].…”

Section: Introductionmentioning

confidence: 99%

Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

Jun¹,

Ray²,

Tolliver³

2020

Preprint

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A classical theorem of Hochster provides purely topological characterization of prime spectra of commutative rings. In this paper, we first prove an analogous statement for idempotent semirings, showing that for a spectral space X, we can construct an idempotent semiring A in such a way that the saturated prime spectrum of A is homeomorphic to X. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring A are spectral, and there is a natural bijection of sets between two. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings.

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Projective systemic modules

Cited by 8 publications

References 33 publications

Vector bundles on tropical schemes

Vector bundles on tropical schemes

$\mathcal{T}$-semiring pairs

Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

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