A nullstellensatz and a positivstellensatz for ordered differential fields

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.     

Alternative evidence for nonnegativity

Sufficient conditions for definiteness of a real polynomial other than the positivstellensatz are defined and illustrated by example.     

Archimedean lattices

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementc∈L, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachx∈L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices. The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.     

Bounded approximation properties in non‐archimedean Banach spaces

Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}$ is not edged?     

Non‐archimedean stratifications of tangent cones

We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.     

Finite union of commutative power joined semigroups

A commutative semigroup is called power joined if for every element a, b there are positive integers m, n such that a^m=bⁿ. A commutative power joined semigroup is archimedean (p. 131, [3]) and cannot be decomposed into the disjoint union of more than one subsemigroup. Every commutative semigroup is uniquely decomposed into the disjoint union of power joined subsemigroups which are called the power joined components. This paper determines the structure of commutative archimedean semigroups which have a finite number of power joined components. The number of power joined components of commutative archimedean semigroups is one or three or infinity. The research for this paper was supported in part by NSF Grant GP-11964.     

阿基米德序半群的链的若干刻画

本文首先引入了一个序半群$S$的准素模糊理想的概念,通过序半群$S$上的一些二元关系以及它的理想的模糊根给出了该序半群是阿基米德序子半群的半格的一些刻画.进一步地借助于序半群$S$的模糊子集对该序半群是阿基米德序子半群的半格进行了刻画.尤其是通过序半群的模糊素根定理证明了序半群$S$是阿基米德序子半群的链当且仅当$S$是阿基米德序子半群的半格且$S$的所有弱完全素模糊理想关于模糊集的包含关系构成链.     

阿基米德序半群的链(英文)

本文通过一个序半群S上的一些二元关系以及它的理想(右理想,双理想)的根集分别给出了该序半群是阿基米德(右阿基米德,t-阿基米德)序子半群的链的刻画.进一步证明了准素序半群是阿基米德序半群的链.最后,通过素根定理证明了序半群S是阿基米德序子半群的链当且仅当S是阿基米德序子半群的半格且S的所有素理想关于集合的包含关系构成链.     

Positive derivations on archimedean lattice-ordered rings

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares. The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.     

Adjoining an Identity to a Reduced Archimedean f-ring, II: Algebras

In “Part I” (presented at Ord05 (Oxford, MS)), we have discussed, for reduced archimedean f-rings, the canonical extension of such a ring, A, to one with identity, uA, and the class U of u-extendable maps (i.e., homomorphisms which lift over the u’s to identity preserving homomorphisms). We showed that U is a category and u becomes a functor from U which is a monoreflection; the maps in U were characterized. This paper addresses the interaction between our functor u, and v , the vector lattice monoreflection in archimedean ?-groups (due to Conrad and Bleier). In short, v restricts to a monoreflection of reduced archimedean f-rings into reduced archimedean f-algebras, ψ ∈ U if and only if v ψ ∈ U, and vu is a monoreflection into reduced archimedean f-algebras with identity. This work was motivated by the question put to us by G. Buskes at Ord05: what maps are o-extendable; i.e., extend over the orthomorphism rings? (The orthomorphism ring oA is a unital extension of uA, and any o-extendable map lies in U.) While a complete answer seems quite complicated (if not hopelessly out of reach), here we shall identify a class of objects D for which oD = vuD and all maps from D lie in U, hence any map from D to a reduced archimedean f-algebra is o-extendable.     

Archimedean ideals of valuation domains

Let R be a valuation domain. We prove that the only archimedean ideals of R are the principal ones or those isomorphic to P and only if the value group of R is complete.     

Zur Theorie der Vollständigen Teilbarkeitshalbgruppen

In this paper we consider complete d-semigroups, i. e. d-semigroups with a conditionally complete lattice satisfying the implication: ⋂a_i ⇒ xsy = ⋂xa.y. As main results we present: 1. Every complete d-semigroup is archimedean. 2. A d-semigroup is complete iff its cone is complete. 3. Every archimedean d-semigroup can be embedded into an archimedean d-semigroup with identity 1. 4. Every complete d-semigroup can be embedded into a complete d-semigroup with identity.      

On Bessel distributions for quasi-split groups

We show that the Bessel distribution attached by Gelfand and Kazhdan and by Shalika to a generic representation of a quasi-split reductive group over a local field is given by a function when it is restricted to the open Bruhat cell. As in the case of the character distribution, this function is real analytic for archimedean fields and locally constant for non-archimidean fields.

     

On the definition of Hilbert space

The projection theorem expresses a central feature of classical Hilbert space. Do other infinite dimensional sesquilinear spaces share this property? We show here that this is not the case for several prominent candidates; in particular Kalish's p-adic Hilbert spaces, Springers non archimedean normed spaces, the positive definite spaces over ordered fields. This yields interesting characterizations of classical Hilbert space.     

序半群的阿基米德半群的半格分解

本文通过一个序半群S上的一些二元关系以及它的理想的根集的性质该序半群是阿基米德半群的半格,特别地是阿基米德半群的链的刻划,证明了S是阿基米德链当且仅当S是准素的.通过序半群的m-系的概念,证明了S的任意半素理想是含它的所有素理想的交,并通过该结论,证明了S是阿基米德半群的链当且仅当S是阿基米德半群的半格且S的所有素理想关于集包含关系构成链.作为应用,该结论在一般的半群（没有序）[1]中也成立.     

Ordered differential fields

We introduce the abstract notion of an ordered differential field and show that some of the basic asymptotic growth properties of differentiable functions can be discussed within this setting. However, in order to ensure that log x → ∞ as x → ∞, we have to assume that the field of constants is archimedean.     

A generalization of a theorem by Bigard–Keimel and Conrad–Diem

Let G be an archimedean ℓ-group and \mathfrakP(G){\mathfrak{P}(G)} denote the set of all polar preserving bounded group endomorphisms of G. Bigard and Keimel in [Bull. Soc. Math. France 97 (1969), 381–398] and, independently, Conrad and Diem in [Illinois J. Math. 15 (1971), 222–240] proved that \mathfrakP(G){\mathfrak{P}(G)} is an archimedean ℓ-group with respect to the pointwise addition and ordering. This classical result is extended in this paper to certain sets of disjointness preserving bounded homomorphisms on archimedean ℓ-groups.