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(1)Title. Riesz空間における,あるBandの構成について. Author(s). 東山, 貞子. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 29(1) : 11-13. Issue Date. 1978-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6019. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section HA) Vol. 29,No. 1 September 1978. Bgft 53 if- 9 ^. On the Construction of a Certain Band in Riesz Spaces. Teiko HIGASHIYAMA Mathematics Laboratory, Kushiro College, Hokkaido University of Education Kushiro 085. XLUl^: Riesz^tc^ttS, ^& Bandc7)^)^^^^T. ^m^K^smw^w^s. Introduction Concerning the paper "On some order properties of Riesz spaces and their relations" by W.. A. J. Luxemburg [I], we consider the construction of the smallest band [A] containing ideal A. A Riesz space is also called a semi-ordered linear space in the terminology of H. NAKANO.. For terminology and notation not explained in this paper we refer to [2]. In what follows L will always denote a Riesz space, i.e., let L be a lattice ordered linear space.. 1. Suprema and infima of directed systems. For a set of element x^L(/(GA) (a) If [x>,'.\^A] is directed upperwards or downwards, then we will write XA^ or x\ 1. HeA Ae/t. respectively. This set, being both directed upperwards and downwards, is called a directed set.. (b) If XA T and sup x^= a, or x». i and inf A;A = a this will be denoted by x\ ^ a orx^i a respectively.. Ae/1. Ai=A. '. /^eA. HeA. Ae/l. ^<=A. We shall show the following relations without proof. (1) {A-A : A£A} consisting of only one element, i.e., Xi=a(A^A) is directed. (2) xn T a^— xi^ —a, XA^ a=>—Xif — a. Ae/1 ^e/1 : ^e/1 ^e/1. (3) Xi/t a, a>0^aXA/t aa, x^i a, a>0=^axn^ aa, AeA. '. \<EA. '. >.eA. >.eA. x\ t a, a<0=>aXA -1- aa, x\ ^ a, a<Q^>ax\\aa. Ae/1. '. .. .. aeA. Ae/1. Ae/1. AeA. .. .. X<=A. (4) ^ T ^=>(.VA+A).T(ff+&),.<;A.^fl^(^+&).-l.(«+A) .. .. ^e/1. .. (5) Xtf'a, yr T ^(.VA+yr).T.(ff_+6),. >,<=A. Ae/1 rer . .^e/l.yer. Xl, 1 a, Vr J- 6=>(-r,<+y/^.l. (a+6). AeA ' ~ /er . ^ AeA, r<=F. (6) x>. T a=> sup (;(;.<, &).T sup (ff, b), ieA. ^<=A. inf (xt, b) T inf(ff, b), A<=A. x i a=> sup (xn, b) i sup (a, b), AeA. '. Ae/1. mf(xi, b) i inf (a, b). >.eA. (11).

(3) S. HIGASHIYAMA. (7) x\ T a, y-f T A=?>sup(^, yy) t sup((Z,A), AeA ~ rer -. ' . ieA,rer. inf(^,y/). T' inf(a,6), ^eA/sr. x^i a,yr 4. &=>sup(^,yy) 4. sup (a, b), AeA ' ~ r<=r . : , . ,te/l,/er. mt(xn,yr). -I- int(a,b). \<=A,7<=r. (8) For^,eL(A£A, r£H) putting/] ={(A, r) : AGA, r^n}, 2== {(&,•••, 5n) •• &,•••, (5n£^, n^N] (where N={0,1,2,---]). Let ^=)S-=(So, ••-, <5n), So=(Ao, ro), •••, Sn=(An, yn). y<s=SUp(x^ro,-",X^rn), Z(f=mf(x>.^ro,'", X^nrn)-. Then we obtain the system ya{a^.2) and za{a^2). %x\ X,X),Y \_x\^>ya i _x, Ae/1 ' rer ~ (Te£. XA -t- X, X\t i_~XA=>Za i X. AeA ' ' rer, " ~ <feZ. 2. Construction of the smallest band [A] containing ideal A (c) A linear subspace A of L is called an ideal A whenever /&A and \g ^ / implies g^A. We have the following results. ( 9 ) L is a band, and {0} is also a band. (10) Let A,i(A<=A) be band, then 0 AA is a band. \eA. By the above results the intersection of all band, which include A, is the smallest band including A. We will denote it by the symbol {A}. Putting B=[a—b '• A^xi, t a, A3y/ T & } where A is an ideal in L. ieA. rer. (11) B={a+b:A=)XAi a, A3yri_b] >,<=A r<=r. ={a-b: A^XA'Ta, A^y.l'b] Proof. xGB=>x=a-b,A^XA .1 a, A~3y-, t_6.. By (2)A3-yr i—b, x= a+( -b) so x^[a+b : A3^.T a, A3y, i _b} there, B^[a +A:A3^.T a,A=)yr l_b]. xe{a+b : A3^.T a, A3yr OH^ff+A, As^.T ff, A3yr ib. 'Since x=b'-(-a), by (2) A^-x^'-a. Consequently, x^[a-b •• A3" yer. XA i ff',-A3yr i_b}.. ,,. ^. e/1. [a+"b: A^xxT a, A=)yr i b}(^{a-b •• A^XA i a, A=)yr 4.&}, x^[a-b: As^l ff, A^y-, i_b}^x=a-b,A^Xi.i a, A-Sy-, I _b. Since x=-b-(-a) by (2) A^-xi.T -a, A^-yr T_-A and so xGB, [a-b: A'3x^i a, A^y-, 4. _b}^-B and we obtain the desired result.. yer. '. Ae/i. r<=r. (12) B=5x^-xGB Proof. x(=B^x=a-b,A^Xi/f a.A^yr T 6. -x=b-aG:B.. (13) A3^.T a=>a(=B,A^x^i a-^aeB. Proof. A^X). T a, for system y,.=0(r<sr)by (1) A3y, T_0, a=a-O^B. And the AeA. '. ~. '. rer. second formula A^—Xnt —a, and so —a^B by (12) aGB.. (14) B=5a> O^AB^.t.ff. Ae/1. (12).

(4) On the Construction of a Certain Band in Riesz Spaces. Proof. If B^a^O, a=c-d, ABMA.T c, A3vy T _d, then for arbitrary but fixed yCF ieA rer. 0^a=c—d<c—Vr,As>(u^—Vr)^ c—Vr putting A3^=inf ((u^—Vr), a) (AGA). i^eA. A^^A.T .inf {(c—Vr), a)=a.. Theorem 1. {A} =B Proof, (i) Ac5:For A3fl, ^(AeA), by (1), (13) A3^ t ff, ffC5. ^e/1. (ii) 5 is a linear space: B3/=(3;—6, A3A-A.T .«, A3.y/t_A. ff>0=> a/ =aa—ab, \e.A yeP. by (3) a/£B,A3ff.VA.T aa,A=^ayr ^ _ab. a<0^- a/e .8, by (12) a/ e^. This shows thatB^f^afeB. Let /, gGB, f=a-b, A3^.t .fl, A=);v, T_A, g=c-d, A^Us T _c. A=)vs^d,f+g=(a+c)-(b+d),A=:i(x,i+us),_^ia+c),A=)(yr+vs) _T. (b+d), 4eJ. and so,f+g<=B.. .-. -. ^eA,ife£. •. yer,. se^l. (hi) B is an ideal: B^f=a—b, AS.C^.T a, A3yy T_A. For arbitrary but fixed y£7-l,. f=a-b^a-yr,f+^(a-y,)+. By (4), (6) A=3(^-y,)+.T .(ff-y,)^A3inf ((^-yr^. /+)^T mt((a-yr)+,f+)=f+. By (13) /+£ B, therefore |/|eB." Assume that 53,, |^|^|/|. By (14ywe have a system ^(A£A) such that A=3x^ T |/|. 0^5-+^|/|, by(6)A3inf (A;A, /+) T mf(\f\,g+)=g+. By (13), ^+eB. On the other hand g~^B because, \-g\=\g\^f,. ,eA. •gY=g~. From linearity of B, g=g —g~G.B.. (iv) B is a band: If 5^/^T /, then 53/,+ T /+, by (14) A^x^r T_/A+(A£/I). Let/|=tl,r);A£yl, yGU} ^=={(^, •••, ^); Sc,---, Sn(=A, nGN}, ^3CT=((5o,'",(5,,) 5o=(Ao, To), •", l5'n=(An, Yn) We have a system A~3ya(a^S) such that y a = sup (x^ro,'", x^rn)- K follows thatA3yo-T. /+. By (13) f+(=B. And by (2), (6) 53/F..I /." Andnow,foranyAe7l,0^/-^/A~£B. Since B is an ideal, f-GB. So f=f+-f~G.B.. (v) B =[A} •• Let X be a band including A. Observe that 53,, f=a- b, A^xnf a, Ae/1. A^y-t T b, ACZX. Therefore we obtain that B ^X and so B is the smallest band including A. rer. The proof is complete.. Theorem 2. If A is an ideal, A is a band <=>{a : A^x^f a}=B. Ae/1. Proof. (-=>) If A is a band then by Theorem 1 Ac{ff : A^^.T ff}c5={A}=A.. (^ ) Let A3/,. T./. /£5=»-/e5={fl:A3^.T.ff}. So, A3^ T_-/, A^-gr i f. >.<=A. .. '. .. .'. ^e/1. '. ~". yer. '. '. For arbitrary AeA, y<=r A^/A^/^—^eA, and hence A is a band. /. ~". yer'. References. [ 1 ] Luxemburg, W. A. J. (1968), On some order properties of Riesz spaces and their relations, Archiv Der Mathematik, Vol. XIX, 488-493. [2] Luxemburg, W. A. J. and Zaanen, A. C. (1971), Riesz spaces. Vol. I North-Holland Math. Library.. (13).

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