Orderings on noncommutative rings

Abstract

Orderings on a noncommutative ring 'A' are defined exactly as in the commutative case. A preordering of 'A' is a subset T ⊆ A such that ΣA²(T) ⊆ T, where ΣA²(T) is the set of all finite sums of permuted products of elements ' a'1,'a'1, ... ,'a n','an ','t'1 , ... ,'t'm for 'a' 1, ... ,'an' ∈ 'A', 't'1, ... ,' tm' ∈ 'T', 'n' ≥ 0, 'm' ≥ 0. A has an ordering if and only if 'A' has a proper preordering (a preordering not containing -1). Given an ordering 'P' of 'A'. A/(P ⋂ — P) is an integral domain. Let Sper('A') be the set of all orderings on 'A'. A version of Positivstellensatz holds for Sper('A'). Sper('A') gives rise to an abstract real spectrum exactly as in the commutative case. For a fixed real primeideal ℘ of 'A', the set of all support ℘ orderings of 'A' forms a space of orderings. Given a noncommutative integral domain 'D', suppose {0} is a real prime ideal of 'D'. Every support {0} ordering of ' D' induces an archimedean equivalence relation on 'D'. This equivalence relation gives rise naturally to a valuation on ' D'. Also, one defines a real place ɑ: (D X D)\(0,0) → ℝ ∪ {∞} associated with this valuation. A version of Baer-Krull theorem holds for this sort of real places on 'D'. There is a natural P-structure on the space of support {0} orderings of 'D'. By allowing valuations in the weak sense, a version of Brocker's trivialization theorem for fans holds. The quantum ring ℝ [x,y] is a special skew polynomial ring, where the multiplication is given by yx=ξxy for some real ξ > 0. ξ ≠ 1. The set of support {0} orderings on the quantum ring is identified with ⋃8i=1Ii (disjoint union), where each 'Ii' is a copy of I=(⋃λ∈ℝFλ) ⋃ {±∞} and Fλ ={λ+λ-} if λ ∈ ℚ and Fλ={λ} if λ ∈ ℝ\ℚ . Here λ+. λ- are just symbols. Let (X, G) be the space of orderings consisting of all support {0} orderings on the quantum ring. The fans of (X, G) are determined completely and the stability index of (X, G) is shown to be 2. The topology on 'X' together with the fans shows (X,G) is indecomposable. The real places on the quantum ring are computed and are shown to be order compatible. Applying results on minimal generation of constructible sets of abstract real spectra to Sper( ℝ[x,y]ξ) some bounds for the number of inequalities to define a basic open, or a basic closed, or a constructible set in Sper(ℝ[x,y]ξ) have been determined.