Non-Archimedean Directed Ordered Vector Spacesの実例について
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 29, No. 2 February 1979. ^mm^±wi: (^ 2 ;;EiiA) ^ 29 ^ ^ 2 ^- Bgffl 54 ^ 2 ^. A Certain Example of Non-Archimedean Directed. Ordered Vector Spaces. Teiko HIGASHIYAMA Mathematics Laboratory, Kushiro College, Hokkaido Univesity of Education Kushiro 085. jf(lijl^rF : Non-Archimedean Directed Ordered Vector Spaces <7)^;f?IJi;:-3^-C. »igl?*'^?^m&TCT%S. 1. Introduction and preliminaries Concerning the paper (1) by W. A. J. Luxemburg, we investigated about the properties of an Archimedean Riesz space.. We recall the definition of an Archimedean Riesz space as presented in (3). Definition. The Riesz space L is called Archimedean, if it is true that -"-% | 0 for all 0^ u^L. n "J.n=i. The lexicographically ordered plane is a well-known example of a non-Archimedean Riesz space. But, in this case it is true that -^-ti\ 0 for some Q^u^L. In paper (3) we offered an n 4.n=i. example of Riesz space in which every ideal is non-Archimedean except {0} . And then, we will. be brief about it. Let Z be a set of all integers, and L is the linear space all functions on Z such that [n ; /(%)+0} is bounded. We introduced a certain order relation in L, therefore L can be a non-Archimdean Riesz space with the conditions mentioned above. In this paper, we want to show the example with the same conditions, defined on R (real numbers). But it will be only a directed ordered vector space, not a lattice.. In what follows, L will denote a directed ordered vector space. The subset P=[x; L^x^O] is called the positive cone of L. P has the following properties. (i) P30, (ii) P=3x, a>Q=>ax(=P, (iii) P^x,y=>x-\-y<=P, (iv) P^x,—x =>x=0, (v)(P)=L i. e.,L to be the linear space generated by P. Con-. versely, if P is a subset of the real linear space L, which holds the properties (i), (ii ),(iii), (iv) and (v), then L is made into a directed ordered vector space by defining that f-^g holds if and only if g—f^P. Evidently, we may just see that L is an ordered vector space. [See (2)]. We will show that L is directed. Let/, g^L. Putting f=pi—p2 and ^i, pz^P, g=q\—qv and 171, q-t^P, then vre have. pi+gi-f=gi+p2<=P^f^Pi+<Si; pi+gi-g=pi+g2<=P=>g^Pi+(ii.. (17).
(3) T. HIGASHIYAMA. This concludes the proof. 2. Example. Let L be the set of all real functions / on (—oo, +00) such that / holds this property (*). (*) SUP (A- ; f(x)^0, —oo<,Z'<+oo}. ForL=i/, we denote that sup (A-;/( A-^O, —°o<x< +<»}==A'/< +°°. Now consider the subset P. of L consisting of all f^-L such that either f(x/) >0 or / =0. And then, P satisfies the conditions of a positive cone. We introduce a partial ordering in L by defining that f^g means that g—f^. P, being a directed ordered vector space with respect to this partial ordering. We shall prove that L is non-Archimedean. For this purpose, we take a real number a<x/ and a function g(x)=l{x=a);g(x)=0(x::fca, -oo<x<+oo) for all 0 < / £ L. Since x\/-g =x/, we have that. (^/-^)(^-.)=^/(^)>0 holds for all n=l, 2, ••• and then -'-f^g>Q for all n=l, 2, ••• by the definition of ordering in L.. n. 1 ,.1°°. Hence, it is false that -±-f | 0. This is a desired result. Especially, for every 0<f<^L, there n~ 4.n=l. exists some element g^L such that nf^g, n=l,2, •••• In fact, taking that a<x/ and g(x)=l (x=a);g(x)=0(x^a, -oo<;r< +00), we have (g-nf} (xg-,,f)=g(a)= 1>0. Since Xg-,,/=a, therefore gt&nf, n=l, 2, •••.. It remains to prove that L is not lattice ordered. We assume that Q,f^g for L^f=f=0, f(xf)=0 implies Xf^Xg. lfxf<Xg,thenputtmgh(x)=l(x=a); h(x)=0 (xzt=a, -oo<x< +00). for Xf <a< Xg, then f <h<g. Consequently g4= sup (/, 0). If there exists g= sup (/, 0) then we obtain that xg=x/, g(xg)>0. Putting h( x )=-^g(xg) (x=xg); h(x)=Q (x^Xg, -°o<.z-<+00).. We also have /< h<g, consequently sup(/, 0)^g. Contradiction. This is complete.. References (1) Luxemburg, W. A. J. (1968), On some order properties of Riesz spaces and their relations. Archiv Der Math. Vol. XIX, 488-493. (2) Nakano, H. (1950), Modern spectral theory. Vol. II, Tokyo Math. Book Series, 23-24. (3) Higashiyama, T. (1978), On the example of non-Archimedean Riesz space. Kushiro Ronshu, No. 10.. (18).
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