Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA local-global principle for preordered semirings and abstract Positivstellensatze
109
0
0.0
(
0
)
Authors
Tobias Fritz
Ask ChatGPT about the research
No Arabic abstract
Motivated by trying to find a new proof of Artins theorem on positive polynomials, we state and prove a Positivstellensatz for preordered semirings in the form of a local-global principle. It relates the given algebraic order on a suitably well-behaved semiring to the geometrical order defined in terms of a probing by homomorphisms to test algebras. We introduce and study the latter as structures intended to capture the behaviour of a semiring element in the infinitesimal neighbourhoods of a real point of the real spectrum. As first applications of our local-global principle, we prove two abstract non-Archimedean Positivstellensatze. The first one is a non-Archimedean generalization of the classical Positivstellensatz of Krivine-Kadison-Dubois, while the second one is deeper. A companion paper will use our second Positivstellensatz to derive an asymptotic classification of random walks on locally compact abelian groups. As an important intermediate result, we develop an abstract Positivstellensatz for preordered semifields which states that a semifield preorder is always the intersection of its total extensions. We also introduce quasiordered rings and develop some of their theory. While these are related to Marshalls $T$-modules, we argue that quasiordered rings offer an improved definition which puts them among the basic objects of study for real algebra.
rate research
Read More
We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.
Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring.
We study algebraic, combinatorial and topological properties of the set of preorders on a group, and the set of valuations on a field. We show strong analogies between these two kinds of sets and develop a dictionary for these ones. Among the results we make a detailed study of the set of preorders on $mathbb Z^n$. We also prove that the set of valuations on a countable field of transcendence degree at least 2 is an ultrametric Cantor set.
In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and anti-bounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.
The multiplicative semigroup $M_n(F)$ of $ntimes n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ 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 $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $mathbb{B}$ and the tropical semiring $mathbb{T}$. In the first case, $M_n(mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(mathbb{T})$ is the semigroup of $ntimes n$ tropical matrices. Ilin has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ 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 $M_n(S)$, 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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
