scholarly journals Tropical ideals

2018 ◽  
Vol 154 (3) ◽  
pp. 640-670 ◽  
Author(s):  
Diane Maclagan ◽  
Felipe Rincón
Keyword(s):  
Special Class ◽  
Tropical Geometry ◽  
Basic Structure ◽  
Chain Condition ◽  
Duke Math ◽  
Polyhedral Complex ◽  
Algebraic Foundation ◽  
Tropical Varieties ◽  
Associated Variety ◽  
Ascending Chain Condition

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

scholarly journals Tropical Ideals

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Diane Maclagan ◽  
Felipe Rincón
Keyword(s):  
Special Class ◽  
Tropical Geometry ◽  
Combinatorial Structure ◽  
International Audience ◽  
Algebraic Foundation

International audience We introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals Tropical geometry and the motivic nearby fiber

2011 ◽  
Vol 148 (1) ◽  
pp. 269-294 ◽  
Author(s):  
Eric Katz ◽  
Alan Stapledon
Keyword(s):  
Euler Characteristic ◽  
Hodge Structure ◽  
Tropical Geometry ◽  
Mixed Hodge Structure ◽  
General Fiber ◽  
Algebraic Torus ◽  
Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

scholarly journals On minimal log discrepancies and kollár components

2021 ◽  
pp. 1-20
Author(s):  
Joaquín Moraga
Keyword(s):  
Blow Up ◽  
Chain Condition ◽  
Fano Varieties ◽  
Finite Sets ◽  
Fixed Dimension ◽  
Log Canonical ◽  
Finite Set ◽  
Canonical Singularities ◽  
Ascending Chain Condition

Abstract In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

Simple Quotients of Euclidean Lie Algebras

1970 ◽  
Vol 22 (4) ◽  
pp. 839-846 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Chain Condition ◽  
Cartan Matrix ◽  
Finite Codimension ◽  
Infinite Dimensional ◽  
Object Of Study ◽  
Ascending Chain Condition

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

scholarly journals On Rational Equivalence in Tropical Geometry

2016 ◽  
Vol 68 (2) ◽  
pp. 241-257 ◽  
Author(s):  
Lars Allermann ◽  
Simon Hampe ◽  
Johannes Rau
Keyword(s):  
Tropical Geometry ◽  
Chow Groups ◽  
Independent Interest ◽  
Main Step ◽  
Rational Equivalence ◽  
Basic Properties ◽  
Boundary Points ◽  
Intersection Ring ◽  
Tropical Varieties

AbstractThis article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

1971 ◽  
Vol 14 (3) ◽  
pp. 443-444 ◽  
Author(s):  
Kwangil Koh ◽  
A. C. Mewborn
Keyword(s):  
Prime Ring ◽  
Chain Condition ◽  
Prime Rings ◽  
Simple Ring ◽  
Descending Chain Condition ◽  
Image Position ◽  
The Right ◽  
Ascending Chain Condition

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

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