Ordered Division Rings Research Articles

Associativity and commutativity of partially ordered rings

Consider a commutative monoid (M,+,0) and a biadditive binary operation μ:M×M→M. We will show that under some additional general assumptions, the operation μ is automatically both associative and commutative. The main additional assumption is localizability of μ, which essentially means that a certain canonical order on M is compatible with adjoining some multiplicative inverses of elements of M. As an application we show that a division ring F is commutative provided that for all a∈F there exists a natural number k such that a−k is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings (“extended” in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e. for (extended) f-rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem.

A solution to the MV-spectrum problem in size aleph one

Denote by IdcG the lattice of all principal ℓ-ideals of an Abelian ℓ-group G. Our main result is the following. TheoremFor every countable Abelian ℓ-group G, every countable completely normal distributive 0-lattice L, and every closed 0-lattice homomorphismφ:IdcG→L, there are a countable Abelian ℓ-group H, an ℓ-homomorphismf:G→H, and a lattice isomorphismι:IdcH→Lsuch thatφ=ι∘Idcf. We record the following consequences of that result:(1)A 0-lattice homomorphism φ:K→L, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Idc, by an ℓ-homomorphism of Abelian ℓ-groups iff it is closed.(2)A distributive 0-lattice D of cardinality at most ℵ1 is isomorphic to some IdcG iff D is completely normal and for all a,b∈D the set {x∈D|a≤b∨x} has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ1. The bound ℵ1 is sharp.Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring.

Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Combining those methods with Baro's Normal Triangulation Theorem, we obtain the following result: TheoremFor every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra. The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field. Theorem For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.

Semilinear stars are contractible

Let $ {\mathcal {R}}$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture of Edmundo et al. (201

The Real Spectrum of a Noncommutative Ring and the Artin-Lang Homomorphism Theorem

Let R be a noncommutative ring. Two epimorphisms $$\alpha_{i}:R\to (D_{i},\leqslant_{i}),\quad i = 1,2 $$ from R to totally ordered division rings are called equivalent if there exists an order-preserving isomorphism ϕ : (D 1, ⩽ 1) → (D 2, ⩽ 2) satisfying ϕ ∘ α 1 = α 2. In this paper we study the real epi-spectrum of R, defined to be the set of all equivalence classes (with respect to this relation) of epimorphisms from R to ordered division rings. We show that it is a spectral space when endowed with a natural topology and prove a variant of the Artin-Lang homomorphism theorem for finitely generated tensor algebras over real closed division rings.

Groups definable in ordered vector spaces over ordered division rings

AbstractLet M = 〈M, +, <, 0, {λ}λЄD〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a ‘definable quotient group’ U/L, for some convex V-definable subgroup U of 〈Mn, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L.

Embedding Ordered Valued Domains into Division Rings

We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings.

Sublattices of Lattices of Convex Subsets of Vector Spaces

Let Co(V) be a lattice of convex subsets of a vector space V over a totally ordered division ring $$\mathbb{F}$$ . We state that every lattice L can be embedded into Co(V), for some space V over $$\mathbb{F}$$ . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional; in this case L also embeds into a finite lower bounded lattice of the form Co(V,Ω)={X⋂Ω | X ∈ Co(V)}, for some finite subset Ω of V. This result yields, in particular, a new universal class of finite lower bounded lattices.

On Ordered Division Rings

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x → xa 2 for nonzero a, instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative case. Further, it is shown that the bounded subring associated with that ordering is a valuation ring which is preserved under conjugation, so one can associate a natural valuation to a semiordering.

Construction of lattice orders on the semigroup ring of a positive rooted monoid

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A [ x ] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d -elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.

On ordered division rings

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin–Schreier) postulates in requiring only the closedness of the domain of

Finitely presented and coherent ordered modules and rings

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features:. (i) Every subring of Q, and every totally ordered division ring, is po-coherent. (ii) For a partially ordered right module Aover a po-coherent poring R Ais po-coherent if and only if Ais a finitely presented .R-module and A +is a finitely generated R +-semimodule. (iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent. (iv) Every finitely po-presented abelian lattice-ordered group is po-coherent.

Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups

AbstractWe prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.Our main result uses constructible sets and can be put in an abstract general framework.Comparison with the standard methods which use forcing (cf. [4]) is sketched.

Weak ∗-Orderings on ∗-Fields

Analogous to the notion of natural valuations of ordered fields, we introduce the notion of order ∗-valuations for any Baer ordered ∗-fields. When the Bear ordered division rings are finite dimensional over their centers, we show that their order ∗-valuations are nontrivial. Using this, we study a new generalization of ∗-orderings, namely, weak ∗-orderings. Unlike ∗-orderings, weak ∗-orderings do exist in Bear ordered ∗-fields odd dimensional over their centers. Moreover, we prove that if the involution is of the first kind, these ∗-fields must be either commutative fields or standard quaternion algebras. Whereas in case the involution is of the second kind, the dimension of these ∗-fields over their centers must be odd. This strong result also implies that the restriction of weak ∗-ordering on any commutative subfield consisting of symmetric elements only is in fact an ordering (not just a semiordering) is these ∗-fields.

A construction of an ordered division ring with a rank one valuation

A construction of an ordered division ring with a rank one valuation

On a generalization of Albert’s theorem

It is proved by A. A. Albert that in an ordered division ring, any element algebraic over the center is central. In this paper, we shall investigate the following problem. LetD be an ordered division ring. Suppose every element inD is left algebraic over a maximal subfieldK. Does it follow thatD=K? We prove that the answers are affirmative in some cases.

Characterizing lineations defined on open subsets of projective spaces over ordered division rings

Characterizing lineations defined on open subsets of projective spaces over ordered division rings

Lattices of Convex Sets

If V is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

Lattices of convex sets

If V is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

Embedding theorems for ascending nilpotent groups

In this paper two proofs will be given showing that an ascending nilpotent group can be embedded in the multiplicative group of a division ring, if and only if its subgroup of elements of finite order can be embedded in a division ring. The first proof will depend on the material in a paper by the author (see [3]), and the second proof will use the results in the first part of the paper On ordered division rings by B. H. Neumann (see [5]). The construction in the first proof is used to show that any automorphism of the group can be extended to the division ring. First we will determine which countably generated locally finite groups can be embedded in a division ring. As a corollary to this a criterion for embedding locally nilpotent periodic groups will be obtained. With this result we will be able to completely determine which ascending nilpotent groups can be embedded in a division ring. In this paper a group G will be said to have property E if it can be embedded in a division ring D, and to have property EE if every automorphism of the group G can be extended to be an automorphism of some division ring D. The theorem on countable locally finite groups depends mainly on a result of Amitsur (see [1]). The following notation is adapted from the notation of Amitsur's paper. 7r will denote the set of all primes and uri the set of all odd primes p such that 2 has odd order mod p. Let m and r be relatively prime integers. Put s = (r -1, m), t = m/s and n = minimal integer satisfying rn_ 1 mod m. Denote by Gm,,r a group generated by two elements A and B satisfying the relations Am= 1, Bn=At, and BAB-l=Ar. Denote by G:,, a group G which has an ascending tower of subgroups 1Hi| 0? i< oo } such that G = U=1 Hi and each Hi is isomorphic to Gmj,ri for some relatively prime integers mi and ri. Let si = (ri -1, m), ti = mi/si and ni = minimal integer satisfying rei-z 1 mod ni. T*, 0* and I* will denote the binary tetrahedral, octahedral, and icosahedral groups (see [1]). Amitsur proved the following: