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A note on finitely generated ideal-simple commutative semirings

Vítězslav Kala, Tomáš Kepka (2008)

Commentationes Mathematicae Universitatis Carolinae

Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.

Algèbres de polynômes tropicaux

Dominique Castella (2013)

Annales mathématiques Blaise Pascal

Nous continuons dans ce second article, l’étude des outils algébrique de l’algèbre de la caractéristique 1 : nous examinons plus spécialement ici les algèbres de polynômes sur un semi-corps idempotent. Ce travail est motivé par le développement de la géométrie tropicale qui apparaît comme étant la géométrie algébrique de l’algèbre tropicale. En fait l’objet algébrique le plus intéressant est l’image de l’algèbre de polynôme dans son semi-corps des fractions. Nous pouvons ainsi retrouver sur les...

Clifford semifields

Mridul K. Sen, Sunil K. Maity, Kar-Ping Shum (2004)

Discussiones Mathematicae - General Algebra and Applications

It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. We have recently extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper, we introduce the notions of Clifford semidomain and Clifford semifield. Some structure theorems for these semirings are obtained.

Idempotent semigroups and tropical algebraic sets

Zur Izhakian, Eugenii Shustin (2012)

Journal of the European Mathematical Society

The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical...

Norms on semirings. I.

Vítězslav Kala, Tomáš Kepka, Petr Němec (2010)

Acta Universitatis Carolinae. Mathematica et Physica

Notes on commutative parasemifields

Vítězslav Kala, Tomáš Kepka, Miroslav Korbelář (2009)

Commentationes Mathematicae Universitatis Carolinae

Parasemifields (i.e., commutative semirings whose multiplicative semigroups are groups) are considered in more detail. We show that if a parasemifield S contains + as a subparasemifield and is generated by + { a } , a S , as a semiring, then S is (as a semiring) not finitely generated.

On ternary semifields

Tapan K. Dutta, Sukhendu Kar (2004)

Discussiones Mathematicae - General Algebra and Applications

In this paper, we introduce the notion of ternary semi-integral domain and ternary semifield and study some of their properties.In particular we also investigate the maximal ideals of the ternary semiring Z¯₀.

Semifields and a theorem of Abhyankar

Vítězslav Kala (2017)

Commentationes Mathematicae Universitatis Carolinae

Abhyankar proved that every field of finite transcendence degree over or over a finite field is a homomorphic image of a subring of the ring of polynomials [ T 1 , , T n ] (for some n depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

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