A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield T, the semifield Z max and the two element semifield B. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. An improvement of a result from [BE13] is proven which can be regarded as a Nullstellensatz for tropical polynomials. 2010 MSC: 14T05 (Primary); 16Y60 (Primary); 12K10 (Secondary); 06F05 (Secondary) Keywords: idempotent semirings, tropical polynomials 2 Prime congruences of semirings In this paper by a semiring we mean a commutative semiring with multiplicative unit, that is a nonempty set R with two binary operations (+, ·) satisfying:(i) (R, +) is a commutative monoid with identity element 0
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many ddimensional toric quiver varieties. A procedure for their classification is outlined.
We describe the class of quiver settings with one dimensional vertices whose semi-simple representations are parametrized by a complete intersection variety. We show that these quivers can be reduced to a one vertex quiver with some combinatorial reduction steps. We also show that this class consists of the quivers from which we can not obtain two specific non complete intersection quivers via contracting strongly connected components and deleting subquivers. We also prove that the class of coregular quivers with arbitrary dimension vector that has been described earlier via reduction steps, can be described by not containing a specific subquiver in the above sense.has been applied in [3] to describe quiver settings with complete intersection quotient varieties (C.I. quiver settings for short) in the special case when the quiver is symmetric and has no loops. The general case however seem to be very difficult to understand from this approach. In this paper we will be concerned with the special case when the values of the dimension vector are all one (but there is no restriction on the structure of the quiver). Though this is a strong restriction from the point of view of representation theory, it still covers a rather interesting class, since the corresponding affine quotients are toric varieties. The question when a toric variety is a complete intersection received considerable attention in the literature, see for example [7] or [8].Our main results are contained in section 6, where we will establish a new reduction-step for quivers with one dimensional vertices and show that all C.I. quiver settings on at least two vertices can be reduced.We will also introduce the notion of descendant, which is similar to graph-theoretic minors (the difference is that only strongly connected components can be contracted) and describe the class in question as the quivers that do not contain certain forbidden descendants. In section 7 we will show that the coregular quiver settings (with arbitrary dimension vectors) can also be described by not containing a single forbidden descendant. These results give some hope that a simlar statement can be formulated in the general case for C.I. quiver settings as well.Throughout this paper we will work over an algebraically closed field of characteristic zero, which will be denoted it by C. This is convenient since we will use several results of Le Bruyn and Procesi [11], and Raf Bocklandt [3,2], who worked with this assumption. However as it was shown by Domokos and Zubkov in [6] many of the results extend to fields with positive characteristic as well. For example the classification of quivers with genuine simple representation we will recall below, holds over an arbitrary field.Acknowledgment: The author would like to thank his supervisor Mátyás Domokos for his invaluable help and guidance in his work. PreliminariesA quiver Q = (V, A, s, t) is a quadruple consisting of a set of vertices V , a set of arrows A, and two maps s, t : A → V which assign to each arrow its starting and terminat...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.