On Ordered Division Rings1
نویسنده
چکیده
1. Ordered groups. Suppose T is a simply ordered set and for each 7 in T let Ry be an additive group. Consider the set H = H(T, Ry) of all vectors ( • • • , ry, ■ • • ) for which ry is in Ry and almost every r"< is zero (i.e., the set of 7's for which ry^0 is inversely well ordered). H is a group w.r.t. (with respect to) vector addition, called the T-sum of the Ry. It the Ry are ordered groups (notation o-group), then H is an o-group w.r.t. the following definition of order: h in H is positive if h 5^0 and the nonzero component with greatest 7 is positive. For the rest of this section let G denote an additive o-group. The set To of all pairs of convex subgroups Gy, Gy of G where Gy covers Gy is simply ordered by inclusion, and each Gy/Gy is o-isomorphic to a subgroup of the additive group of real numbers. Thus G determines a TG-sum H = H(Tq, Gy/Gy), and H is an abelian o-group. The value 7(g) of g?^0 in G is the 7 in To for which g is in Gy but not in Gy. If a and b are elements of H for which by = ay for all 7>|3, by = 0 for all 7 ?£p\ and o^O, then 0 is called the $th head of a.
منابع مشابه
FUZZY FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN PARTIALLY ORDERED METRIC SPACES
In this paper, we consider fuzzy fractional partial differential equations under Caputo generalized Hukuhara differentiability. Some new results on the existence and uniqueness of two types of fuzzy solutions are studied via weakly contractive mapping in the partially ordered metric space. Some application examples are presented to illustrate our main results.
متن کاملDirect synthesis of partially ordered tetragonally structured FePt nanoparticles by polyol method for biomedical application
We report the direct soft chemical synthesis and characterization of a family of face centered cubic (fcc) and partially ordered face centered tetragonal (fct) FePt nanoparticles (NPs) suitable for biomedical applications. Both fcc and partially ordered fct-FePt NPs are synthesized by employing a simple polyol method. By using polyvinyl pyrolidone (PVP) as a stabilizer in various ratios during ...
متن کاملDirect synthesis of partially ordered tetragonally structured FePt nanoparticles by polyol method for biomedical application
We report the direct soft chemical synthesis and characterization of a family of face centered cubic (fcc) and partially ordered face centered tetragonal (fct) FePt nanoparticles (NPs) suitable for biomedical applications. Both fcc and partially ordered fct-FePt NPs are synthesized by employing a simple polyol method. By using polyvinyl pyrolidone (PVP) as a stabilizer in various ratios during ...
متن کاملCENTRAL EXTENSIONS OF n–ORDERED DIVISION RINGS
We study central extensions of division rings with orderings of higher level. We show that orderings extend to certain immediate and inert extensions. Further, it is proved that any n–ordered division ring can be extended to a n–ordered division ring with almost real closed center. In the last section an old theorem due to Neumann is generalized: every n–ordered division ring can be extended to...
متن کامل