Algebras with a negation map

Rowen, Louis

doi:10.1007/s40879-021-00499-0

Cited by 9 publications

(12 citation statements)

References 39 publications

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“…In [Row19,Row22], Rowen introduced the notion of systems and in particular of semiring systems, which appeared to be more general than the notion of hyperfield as shown in [Row22,AGR22]. We recall the definition, since we shall use this notion later.…”

Section: 3mentioning

confidence: 99%

“…In [Row19,Row22], Rowen introduced the notion of semiring system which involves a subset of tangible elements and a surpassing relation, and in [AGR22], it is shown that under some conditions, the set of tangible or zero elements of a semiring system is isomorphic to a hyperfield, and that any hyperfield can be written as the set of tangible or zero elements of a semiring system. In particular, the semiring system associated to the supertropical semiring [IR10] is such that the set of tangible or zero elements coincides with R max , and is isomorphic to the tropical hyperfield [Kra83,Vir10,CC11a].…”

Section: Introductionmentioning

confidence: 99%

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Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

Akian,

Gaubert,

Tavakolipour

2023

Preprint

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The symmetrized tropical semiring is an extension of the tropical semifield, initially introduced to solve tropical linear systems using Cramer's rule. It is equivalent to the real tropical hyperfield, which has been used in the study of tropicalizations of semialgebraic sets. Polynomials over the symmetrized tropical semiring, and their factorizations, were considered by Quadrat. Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots of univariate polynomials over hyperfields. In the special case of the hyperfield of signs, they related multiplicities with Descarte's rule of sign for real polynomials. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them with the multiplicities of roots over these semirings. We deduce a Descartes' rule for "signs and valuations", which applies to polynomials over a real closed field with a convex valuation and an arbitrary (divisible) value group. We show in particular that the inequality of the Descartes' rule is tight when the value group is non-trivial. This extends to arbitrary value groups a characterization of Gunn in the rank one case, answering also to the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

Akian,

Gaubert,

Tavakolipour

2023

Preprint

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“…In other words, it may be considered as one more stage of a wider program initiated some years ago by the second author, through the theory of triples and systems (see e.g. [21,22]), which has already proved successful in revisiting classical algebraic phenomena by embedding them in a tropical context. Among its applications, we recall the construction of an effective tropical substitute of the exterior algebra, along with a natural extension of the Cayley-Hamilton theorem [11] for endomorphisms of modules over semialgebras.…”

Section: Introductionmentioning

confidence: 99%

Lie pairs

Gatto,

Rowen

2024

Communications in Mathematics

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Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.

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“…A hyperfield is a field-like structure where the latter property is relaxed for the additive operation. In the literature, such structures appear perhaps more than one would expect: hyperfields are of interest, e.g., in tropical geometry [1][2][3], symmetrization [4][5][6], projective geometry [7], valuation theory [8][9][10][11], and ordered algebra [12][13][14]. There are even reasons to believe that their theory generalizes field theory in ways that can be used to tackle deep problems such as the description of F 1 , the "field of characteristic one" (cf.…”

Section: Introductionmentioning

confidence: 99%

A Result of Krasner in Categorial Form

Linzi

2023

Mathematics

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In 1957, M. Krasner described a complete valued field (K,v) as the inverse limit of a system of certain structures, called hyperfields, associated with (K,v). We put this result in purely category-theoretic terms by translating it into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms. We replace the original metric-dependent arguments employed by Krasner with a clean and elegant transition to certain slice categories.

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Algebras with a negation map

Cited by 9 publications

References 39 publications

Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

Lie pairs

A Result of Krasner in Categorial Form

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