On the structure of semifields and lattice-ordered groups

Weinert, Hanns Joachim; Wiegandt, R.

doi:10.1007/bf01879738

Cited by 13 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(In fact, every idempotent semifield arises in this way -see Lemma 2.1 below.) For ease of reference we record (with no claim of originality) a number of facts about semifields, which have been observed by many authors (see for example [15,19,44]): Lemma 2.1. Let S be a semifield.…”

Section: 1mentioning

confidence: 99%

Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

View full text Add to dashboard Cite

The multiplicative semigroup Mn(F ) of n × n matrices over a field F is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form Mn(S), where S is a semiring, and the subsemigroups U Tn(S) and Un(S) of Mn(S) consisting of upper triangular and unitriangular matrices. Our main interest is in the case where S is an idempotent semifield, where we also consider the subsemigroups U Tn(S * ) and Un(S * ) consisting of those matrices of U Tn(S) and Un(S) having all elements on and above the leading diagonal non-zero. Our guiding examples of such S are the 2-element Boolean semiring B and the tropical semiring T. In the first case, Mn(B) is isomorphic to the semigroup of binary relations on an n-element set, and in the second, Mn(T) is the semigroup of n × n tropical matrices.It is well known that the subsemigroup of Mn(F ) consisting of the singular matrices is idempotent generated. We begin consideration of the analogous questions for Mn(S) and its subsemigroups. In particular we show that the idempotent generated subsemigroup of U Tn(T * ) is precisely the semigroup Un(T * ).Il'in has proved that for any semiring R and n > 2, the semigroup Mn(R) is regular if and only if R is a regular ring. We therefore base our investigations for Mn(S) and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of Mn(S), including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.

show abstract

Section: 1mentioning

confidence: 99%

Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

View full text Add to dashboard Cite

show abstract

“…Put another way, although the polynomial semiring † is not a semifield † , it is a cancellative semiring † when its elements are viewed as functions, so we can view polynomials as functions and then pass to the semifield † of fractions, which is called the semifield † of rational functions over F . (The information thereby lost is compensated by Homomorphisms of idempotent semifields † have been studied long ago in the literature [12,39,38], where the homomorphic images are described in terms of what they call (semifield † ) kernels, which as noted in [12] are just the convex (normal) l-subgroups of the corresponding lattice-ordered groups. Since semifield † kernels are subgroups which can be described algebraically, cf.…”

Section: Introductionmentioning

confidence: 99%

Kernels in tropical geometry and a Jordan–Hölder theorem

Perri

Rowen

2018

J. Algebra Appl.

View full text Add to dashboard Cite

Abstract. A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic geometry. In this context the natural analog of the polynomial ring over a field is the polynomial semiring over a semifield, but one obtains homomorphic images of coordinate algebras via congruences rather than ideals, which complicates the algebraic theory considerably.In this paper, we pass to the semifield F (λ 1 , . . . , λn) of fractions of the polynomial semiring, for which there already exists a well developed theory of kernels, also known as lattice ordered subgroups; this approach enables us to switch the structural roles of addition and multiplication and makes available much of the extensive theory of chains of homomorphisms of groups. The parallel of the zero set now is the 1-set.(Idempotent semifields correspond to lattice ordered groups, and the kernels to normal convex l-subgroups.)These notions are refined in the language of supertropical algebra to ν-kernels and 1 ν -sets, lending more precision to tropical varieties when viewed as sets of common roots of polynomials. The ν-kernels corresponding to (supertropical) hypersurfaces are the 1 ν -sets of corner internal rational functions. The ν-kernels corresponding to "usual" tropical geometry are the regular, corner-internal ν-kernels.In analogy to Hilbert's celebrated Nullstellensatz which provides a correspondence between radical ideals and zero sets, we develop a correspondence between 1 ν -sets and a well-studied class of ν-kernels of the rational semifield called polars, originating from the theory of lattice-ordered groups. This correspondence becomes simpler and more applicable when restricted to a special kind of ν-kernel, called principal, intersected with the ν-kernel generated by F . We utilize this theory to study tropical roots of finite sets of tropical polynomials.For our main application, we develop algebraic notions such as composition series and convexity degree, along with notions having a geometric interpretation, like reducibility and hyperdimension, leading to a tropical version of the Jordan-Hölder theorem for the relevant class of ν-kernels. IntroductionThis paper is a combination of [29] and [30]. The underlying motive of tropical algebra is that the valuation group of the order valuation (and related valuations) on the Puiseux series field is the ordered group (R, +) or (Q, +) (depending on which set one uses for powers in the Puiseux series), which can also be viewed as the max-plus algebra on (R, +) or (Q, +). This leads one to a procedure of tropicalization, based on valuations of Puiseux series, which takes us from polynomials over Puiseux series to "tropical" polynomials over the max-plus algebra. One of the main goals of tropical geometry is to study the ensuing varieties. Traditionally, following Zariski, in the affine case, one would pass to the ideal of the polynomial alge...

show abstract

“…First, assume that T is a chain. Then ≤ G(T,v0) is defined as the lexicographical ordering on G(T, By the well-known correspondence of ℓ-groups and commutative additively idempotent parasemifields, G(T, v 0 ) can be treated as an additively idempotent parasemifield and ≤ G(T,v0) is the natural ordering on this parasemifield (for example see [36,37]). Proposition 3.4.…”

Section: Every Monoid Associated To a Finite Tuple Of Semiring-generamentioning

confidence: 99%

Idempotence of finitely generated commutative semifields

Kala

Korbelář

2018

Forum Mathematicum

View full text Add to dashboard Cite

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.2010 Mathematics Subject Classification. primary: 12K10, 16Y60, 20M14, secondary: 11H06, 52A20.This statement can be now understood as an extension of the folklore theorem referred in the beginning, i.e., that every (commutative) field that is finitely generated as a ring is finite. Of course, the greatest difference is the existence of additively idempotent semifields.Since every semifield is clearly ideal-simple, part (b) of the theorem follows immediately from (c). Also, one can be slightly more precise in the additively constant case: this occurs if and only if there is a finitely generated (multiplicative) abelian group G(·) and the semiring is the semifield S := G ∪ {o}, where o is a new element. Operations that extend the multiplication on G are defined by a+b = o and a·o = o for all a, b ∈ S.

show abstract

On the structure of semifields and lattice-ordered groups

Cited by 13 publications

References 5 publications

Matrix semigroups over semirings

Matrix semigroups over semirings

Kernels in tropical geometry and a Jordan–Hölder theorem

Idempotence of finitely generated commutative semifields

Contact Info

Product

Resources

About