MINI [ v \ ‘M H I I I PRIME EDEALS IN A VECTOR LATTICE AND {TS .DEDEKiND COMPLETlON 533°: WWI/WWW”!IWWUHJMW Thesis far the Degree caf Ph. D. MICHiGAH STATE UMVERSITY K. KUMARAN KEITH 1970 LIBRARY + Michigan State. ‘ University ‘ THES‘S This is to certify that the thesis entitled Prune $60.19 leak Vedas» Ldfa. we fix Mmacwsw presented I»; has been accepted towards fulfillment of the requirements for mdegree inMfl‘Qflj’.‘ " ajor professor mew 0-169 LIBRARY muoens 7 ABSTRACT PRIME IDEALS IN A VECTOR LATTICE AND ITS DEDEKIND COMPLETION BY K. Kumaran Kutty Structure spaces are used to study vector lattices by representing a vector lattice as a class of functions on its structure spaces. Masterson has attempted to study the relationship between the structure spaces of an Archimedean vector lattice E and its Dedekind completion E. He has proved that, in the presence of P-P-, starting with a prime ideal P in B we can get a prime ideal 3 in E such that S n E = P. We prove that if E is an arbi- trary Archimedean vector lattice, starting with a prime ideal P in E. wecangetaprime ideal Q in E9 QHE=P.‘ Masterson introduces the property: V x E El', the exis— tence of y E E+ and scalar a such that ay g_x g_y.. and points out that in the presence of this property, the structure spaces of E and E are homeomorphic. ‘We Obtain several conditions equivalent to this property and prove that this pr0perty is strictly stronger than P-P° It is well known that if 0 < x g_y1 + y2., 'where x, y1 y2 6 E, K. Kumaran Kutty then x can be decomposed as x1 + x2 where O g_x1 g_y1 and 0‘s x2.g yz. If x 6 E and y1, 'y2 E E, it is not necessarily true that x1 6 E,. x2 €.E. If every x can be decomposed this way for all yl. y2 E E, starting with a prime ideal P in E., we can Obtain a prime ideal E, in E such that 3'0 E = P. This prOperty is strictly stronger than P'P° and weaker than the property introduced by Masterson. Also we Obtain a way of Characterizing this pr0perty using prime ideals. If g(P) = B; then $ and 9($) are homeomorphic where $ is the structure space of all prime ideals of E. We Obtain two ways of characterizing P-P-, one by a property analogous to the one introduced.by Masterson, and one by a property analogous to the one mentioned above. Also we mention a property which is strictly stronger than the one introduced by Masterson. PRIME IDEALS IN A VECTOR LATTICE AND ITS DEDEKIND COMPLETION BY K. Kumaran Kutty A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1970 G— @55qu / "90~ ’7/ Acknowledgement My thanks are due to Professor John J. Masterson for suggesting this area of research and for the discus- sions I had with him. I wish especially to acknowledge his sustaining encouragement in the initial stages of my work. I am grateful to Mrs. G. Milligan for doing an excellent job of the typing. ii Table of Contents Page Introduction iv Section 0 1 Section 1 6 Section 2 21 Section 3 30 Bibliography 38 iii Introduction Several authors have attempted to study vector lattices using prime ideals. Yosida (8), using prime ideals, proved that every Archimedean vector lattice is isomorphic to a vector lattice of extended functions on some locally compact Hausdorff space. Nakano (7) has proved that every Dedekind U-complete vector lattice E Sis isomorphic to a vector lattice of extended functions on some totally disconnected Hausdorff space X. The space X is Obtained by providing the collection of all maximal dual ideals in the distributive lattice of projectors on E with the dual hull-kernel topology. JOhnson and Kist (2) have shown that the representations of Yosida and Nakano can each be Obtained by considering a suitable subspace of the space of all prime ideals. They have generalized Nakano's representation to arbitrary Archimedean vector lattices. Using the concept of spectral function, Amemiya (1) has developed a spectral theory for vector lattices, generalizing Nakano's theory for the o-complete and complete cases. Johnson and Kist (2) have shown that Amemiya's theory can be Obtained by ideal - theoritic methods. They do this by showing that the set of all spectral functions defined on a vector lattice E is essentially the same as the set of all prime ideals in E. iv The purpose of this thesis is to investigate 0 the relationship between structure spaces of an Archimedean vector lattice E and those of its Dedekind completion A E . Masterson (5) has attempted to answer this question, but his answers are incomplete. He has shown that if E has projection prOperty, for every prime ideal P in E A A A there exists a prime ideal P in E such that P n E A P is minimal prime if P is minimal prime. If m is a ll "U ' structure space of E consisting entirely of minimal prime ideals, 91 and f(fll) are homeomorphic where f(P) = 9.. He also Obtains several conditions equivalent to the homeomorphism of 8 and f(m), where 8 is any structure space of E, in the presence of projection property. He introduces a pr0perty in a vector lattice, viz. for every x 6 El. the existence of y 6 E+ and positive scalar on a ay 3 x g y , and shows that under this condition B and f(m) are homeomorphic where Q is any structure space of E. We show that if P is any prime ideal in an Archimidean vector lattice E., there exists a prime ideal IQ in E such that Q n E = P . This prime Q is not unique. We prove two partial converses to Masterson's result: (1) if P is prime in E for every prime P in A E, then E has P-P-P- (2) If every prime in E is of A the form P for some prime P in E. then E has P-P- Also the property introduced by Masterson is equivalent ~ A to the prOperty that every prime ideal in E is of the form 9 for some prime P in E.. We Obtain other equivalent conditions to this property. we give examples to show that the implications in (l) and (2) above are not reversible. Whether projection property is strictly stronger than the property that P is prime in E for every prime P in E, is an Open question. We introduce another property: if 0 < x g_yl + y2, where x 6 E+, 'y1, y2 6 3+' the decomposability of x as = x1 + x2 where 0.3 xl'S-Yl and 0 3.x2 g_y2 where x1, x2 6 E. It is well-known that if x1 yl, y2 are positive elements of a vector lattice E such that x g.y1 + y2, then x can be expressed as = x1 + x2 where O g_xl g_y1 and O g_x2 g_y2 . This prOperty is known as the Riesz dominated decomposition property. But if x E E+ and yl, y2 E El', it is not necessarily true that 1, x2 E E. In the presence of this property, starting with a prime ideal P in El, we can define a prime ideal is X A ~ ~ A in E :} P H E = P. Also if P is prime in E for all prime P in B, then E has this property. If u is any structure space of E.. 3 and f(fi) are homeomorphic under the mapping f(P) = P . we show that the above mentioned property is strictly stronger than P-P- The property vii introduced by Masterson is stronger than this property, but it is not known whether it is strictly stronger. We also introduce a prOperty strictly stronger than the . A one introduced by Masterson, viz. for every x1. x 6 E+ 2 such that x1 < x2, the existence of y 6 E+ such that x1 < y < x2. The whole work is divided into four sections. Section 0 consists of definitions, etc. necessary for the understanding of the text. Sections 1, 2, 3, constitute the main body of the work. Section 0 Notation 0.1: E+ = {x c E : xig 0]. Definition 0.2: E is Archimedean iff x,y E E+ such that nx g_y for all positive integer n = x = 0, for all x,y. Definition 0.3: E is Dedekind Complete iff [x ] c E, x < y P E V a = sup x exists in E. a d- a a Definition 0.4: If E and E' are vector lattices a mapping ¢ : E ~ E' is a vector lattice homomorphism iff (i) ¢(aa) = a¢(a) for all a e E, for all scalar a. (ii) ¢(a v b) = ¢(a) v mm for all a,b e E. Definition 0.5: A vector lattice isomorphism is a one-one vector lattice homomorphism. Definition 0.6: The vector lattices E and E' are said to be isomorphic if there exists a vector lattice isomorphism ¢ : E d E' such that ¢ is onto. Definition 0.7: E' C E is said to be order dense 1n E if for all x c E, there ex1sts {Xa}aEA c E such that x = sup xa. a Theorem 0.8: (Nakano, 6). If E is a vector lattice, there exists a Dedekind Complete vector lattice A E such that E can be imbedded as an order dense sub- A A vector lattice of E, iff E is Archimedean. If E exists, it is unique upto vector lattice isomorphism. Property 0.9: (Riesz) If 0 < x < y1 + Y2: x,y1.y2 E E, there exists a decomposition of x such that x = + x where 0 < x1 < Y1 and 0 < x2 < y2. x1 2' This prOperty is known as Riesz dominated decomposition property. Definition 0.10: A subset E' C E is order 0 c ' O 0 closed 1ff [xd}a€A E , 52p Xa ex1sts in E = sup XO 6 E'. a Definition 0.11: A linear subspace I c E is an ideal iff I is solid, i.e. x E I, Iyl 3 IX] = y 6 I. Definition 0.12: A principal ideal is an ideal generated by a single element. Equivalently, an ideal I is principal iff there exists a E E 9 I = [x:lx| g nlal, for some integer n}. Definition 0.13: A band is an order closed ideal. Definition 0.14: A principal band is the order closure of a principal ideal. Notation 0.15: For all x F E, x+ = x v 0 X I! >'< < O IX! = X+ V X- = x+ + x- in“ Definition 0.16: X l y iff 'x' A IYI = 0. Notation 0.17: If A c.E, Al = [x c E: x l a In for all a E A]. Theorem 0.18: AL is a band for any subset A of E. Theorem 0.19: E is Archimedean iff A11 = A, for every hand A in E. Definition 0.20: E has Projection Property (P.P.) iff E = B @>B*. for any band B of E. Definition 0.21: E has Principal Projection Property (P.P.P.) iff E = B m BL, for any principal band B of E. Definition 0.22: E is Dedekind O-Complete iff {x l n nFN' xn 3.x 5 E = SUP {Xn} ex1sts in E. n Theorem 0.23: E has P.P. : E Ded. Complete g E has P.P.P. = E § 0 o E is Ded. O-Complete ls ArChlnedean None of the above implications is reversible. Projection Property and Dedekind O-Completeness are independent. If E is Dedekind O-Complete and has P.P. it is Dedekind' Complete. Definition 0.24: An ideal P is prime e x A y C P x,y G E = x 6 P or y 6 P. Theorem 0.25: (Johnson & Kist, 2): The following are equivalent: (1) P is a prime ideal. (2) If x A y = 0, then x 6 P or y 6 P. (3) The quotient vector lattice % is linearly ordered. (4) If P D A 0 B, where A and B are ideals in E, then either P D A or P 3 B. Definition 0.26: Let h denote the collection of all prime ideals in E. For $' C T, the kernel Of $' = “{P : P 6 T'}. The kernel is an ideal, not necessarily prime. The hull of an ideal is the collection of all prime ideals containing that ideal. Hence if $' c T, note that h(k($')) D T', where h denotes the hull and k, the kernel. Taking h(k($')) as the closure of h', the closure of any subset of T is uniquely defined. And this closure Operation defines a topology on T. This topology is known as the hull-kernel topology. T, with this topology is known as the structurespace of E. The class {halaCE is a base for this topology, where Ta = [P E T : a f P}. More generally, let T denote any collection of prime ideals in E such that “(P . P 6 T] = 0. Define the topology on T as above. T, with this topology is a structure space of E. Section 1 It is well known that in a vector lattice, maximal ideals are prime. Yosida (8) has generalized this result. He has shown that relatively maximal ideals are prime. An ideal is relatively maximal if it is maximal with respect to not containing a fixed element. I). y, In this section we given an alternative proof of Yosida's result (Theorem 1.1). We also give some generalizations of his result (Theorem 1.5, Theorem 1.9). We obtain many beautiful results on prime ideals, earlier obtained by Johnson and Kist (Johnson & Kist 2) as easy corollaries to these theorems (Corollaries 1.3, l.4,l.7,1.8). Masterson (5) has shown that starting with a prime ideal P in an Archimedean vector lattice E, an ideal P can be defined in E, the Dedekind Completion of BE, such that P H E = P. He shows that if E has P.P., P is prime in E. We show that for a prime ideal P in an Archimedean vector lattice E, there exists a prime ideal Q in E such that Q n E = P (Theorem 1.10). This prime ideal Q is not unique (Example 1.11). We show that if P is prime in E for every prime P in E, E has P.P.P. (Theorem 1.12). Also this property is strictly stronger than P.P.P. (Example 1.13). We give an example to show that Dedekind O-Completeness, a property stronger than P.P.P., is not strong enough to imply this property (Example 1.14). Recall Masterson's result that P.P. implies this property. Whether P.P. is strictly stronger than this prOperty is not known. Hence we have the chain A P.P. = P prime in E for every prime = P.P.P. & P in E 4 Bed. 0 Completeness Theorem 1.1: Let E be a vector lattice and a E E. Let I be an ideal in E, maximal with respect to not containing a. Then I is prime. Proof: Existence by Zorn's Lemma: Let 3a = (I : I an ideal in E, a f I}. We can take a positive for, a E I e la! 6 I. 3a is non—empty for, 3a empty = every ideal contains a = E has only one non-trivial ideal = E_ is linearly ordered. Ra is partially ordered by inclusion. If II C 12 C... is a chain in q ,U Ii is an upper bound for this chain. .1 ‘a 3a has a maximal element, by Zorn's lemma. Let M be a maximal element. we show that M is prime. Suppose not. Then S'b. C E E.‘b A C = 0. and b )6 M, c E M. = a E (M,b) and a E (M,c) where (M,b) denotes the smallest ideal containing all the 8 elements of 1M and b etc. But a 3 Am + Bb and a g A'm' + B’c where A,B,A',B' are positive scalars I and m, m 6M. By Riesz dominated decomposition property (0.9), a=a1+a2 where ogalgAm 0.3 a2 g_Bb I I I I a = a1 + a2 where 0.3 a1.g A'm 0 g,aé g_B’c I I a2 g_a — a1 + a2 . a2 = a21 + a22, where 0.3 a21 3 al I OSazz-gaz b A c = 0 = a2 A a2 = 0 = a22 = 0 . I .. a a21 g a1 . __ I II .. a — a1 + a2 g_a1 + a1 g_Am + A m = a 6 M , contradiction. Corollary 1.2: The intersection of all relatively maximal ideals in a vector lattice is [0}. The following two results were Obtained by Johnson and Kist(2). They are immediate from the above theorem. Corollary_1.3: The intersection of all prime ideals in a vector lattice is (0] . Corollary_l.4: The intersection of all minimal prime ideals in a vector lattice is {0}. Theorem 1.5: Let I be an ideal in E and O < a f I. Let M be maximal in the class of ideals containing I and not containing a. Then M is prime. Proof: Existence by Zorn's Lemma: Let JIa=[J: J an ideal in E,JDI, azJ} .1 I JI,a is non—empty, for I E JI,a . JI,a is partially ordered by inclusion. Let J1 C J2 C be a chain in .. J - U J. is an upper bound for this chain.'.°J has a La 1 La maximal element, by Zorn's Lemma. Let M be a maximal element. We show that M is prime. Suppose not. =3b,cEE, bAc=0,b£M, CEM = a E (M,b) and a E (M,c) => agAm+Bb, agA'm'+B'c I where A, B, A', B' are sitive scalars and m, m E M . By Riesz dominated decomposition property a=al+a2 where 0_<_al_gAm Cgangb a=a£+ az’ where ogaiSA'm' 0_<_a2'_<_B'c a2_<_a=ai+a2' .. a = a21 + a22 where 0 3 a21 3 al . _ I I I .. a — a1 + a2 g_a1 + a1 g_Am + A m = a E M.. Contradiction. Definition 1.6 (Johnson and Kist): A prime ideal belonging to an ideal I is a prime ideal containing the ideal I. A minimal prime ideal belonging to an ideal I is a minimal element in the class of prime ideals belonging to I. The following two results were obtained by Johnson and Kist. (Johnson and Kist, 2). They follow immediately from the above theorem. Corollary 1.7: The intersection of all prime ideals belonging to an ideal I is I. Corollary 1.8: The intersection of all minimal prime ideals belonging to an ideal I is I. + Theorem 1.9: Let {xa}a€A C E such that x A x A °'°°° A x > 0,. for all integers k and a1 dz Ok for all choices of indices a1, a2,---, dk €.A.. Let M be a maximal element in the class of ideals not containing any of these finite infima. Then M is prime. Proof: Suppose not. Then 3 a,b E E, a A'b = 0 and a EM, b £M.Hence 51 indices a1, 0L2,---, a and ll \‘4 Bl: 620 °°'o B x A x A °°°° A x E (M,a) m a1 a2 an XB AxE3 A°°°°Ame€(M,b) l 2 = Xa A xa A -:-- A Xan g_Am + Ba 1 2 I xB A xB A -- A me g_Cm + Db 1 2 where A, B, C, D are positive scalars and m, m' €:M . So, }{ A x A ---° A x A x A ---- A x d1 d2 an 61 8m _<_ (Am + Ba)A(Cm’ + Db) g (Am+Ba)ACm’ +(Am+Ba)ADb gAmACm’+BaACm’+AmADb+Ba/\Db .3 Am A Cm’ + Cm' + Amw, '3 a A'b = O =9 O 0,, for all integers k and all 0.1 (12 Gk choices of indices dl, d2,..., ak,. for if at least one such infimum is 0 at least one Xa is in P. For the same reason, this class is closed under finite irJ fima o 13 Consider the class 3 = [J an ideal in E: xa 2’? for all a,. J D P}. The class 3 is non-empty for P E 3 where Q = [y E E: IyI g_IxL, for some x in P} and 3 is an ideal in E. 3 is partially ordered by inclusion. If Jl C J2 C°°~~ is a chain in 3, U Ji an upper bound for this chain. .1 the class 3 has a maximal element, by Zorn's lemma. Let Q be a maximal element in 3..‘We show that Q is prime in E. Suppose not- Then 3 aib 6 E a a A'b = O and a 2'0.. b E'Q = 3 a 6 A a xa g_Aq + Ba and 386A 7. ng’+Db x B where A, B, C, D are positive scalars, and l Osq.q 60 = O < xa A x6 3 (Aq + Ba)A(Cql + Db) _gAchq’+AqADb+BaAcq’+BaADb gAqACq’+Aq+Cq’. aAb=0 ='0 < xa A xB E Q,, contradiction. Q n E = P,, by construction. The following example shows that the prime ideal Q obtained in the above theorem is not unique, for a given P in E. Example 1.11: Let E = C, the space of real A convergent sequences, E = m,. the space of real bounded 0' the space of zero convergent sequences is A A a prime ideal in C, but CO is not prime in E, sequences. C 14 A where CO = [x E m: [XI 3 y,. for some y in A CO). CO is an ideal in m. By Corollary 1.7 there is A more than one prime ideal in m containing C0" Let g be any for, I I O A if there IS only one, it has to be CO' one of them» Then q n C 3 C But q n C = C for, 0 ' 0' if not,a sequence converging to a non—zero number is in q = 1 E q = q = m. In fact we prove that there are an Iinfinite number of choices for q. Let a = (O O " 0 1 O --- O 1 O °°°° O 1 0 "'° ) d1 a2 a3 be an element of m,, which has 1's in the positions a1, a2, --° and 0’s at N - {A} where N denotes the positive integers and A = [a1]; . Let 38 a = (I: 0' A I an ideal in m, I:Co,a E I}. This class is non—empty for A C is in it. By Zorn's lemma, this class has a maximal 0 element. Let qa be one such. By methods similar to the ones employed in the proof of Theorem 1.10, we can show that qa is prime in m. Also a' E qa where a’ is the element in m with zeros at A and 1's at N - A , since a' A a = 0 . = -there are at least two prime ideals viz. qa and q;;qanC=C qénC=C O: O I In fact there are an infinite number of choices for qa. 'by lemma 2.4. For we can have a sequence al < a2 < "'°° , where the inequalities are strict 15 and each ai is an element of the type a. By lemma 2.4, Ra = Rb a Ia = Ib where Ra" Rb denote the class of relatively maximal ideals not containing a, b and I , I a denote the principal ideals generated b by a,b. =R CR CR a171a2¥a3 Masterson (5) has proved that if E has P°P-, A and if P is a prime ideal in E, P is a prime ideal . A A A . in E , where P = {y 6 E : IyI _<_ |x| , for some x in A A P]. We prove below that if P is prime in E for all prime P in E, then E has P'P-P- The example given after the Theorem shows that the converse implication is not true. Theorem 1.12: Let E be an Archimedean vector lattice and P a prime ideal in B. Let A A A P = {y 6 E : IyI g_lxl, for some x in P}. If P A is prime in E, for all prime P in E, then E has P'P°P- Proof: Suppose E does not have P-P-P- Then 3 a, b E E+ such that sup (b A na) does not exist in E. n A By theorem 0.23, 5gp (b A na) exists in E. Let this sup be denoted by a(b). A .2 b = a(b) + b - a(b) in E. A Consider the class 3 = {I an ideal in E: x E E. le < a(b) = x 6 I: a(b) Z I}. This class is l6 non—empty, for, the ideal generated by all elements of E less than a(b) is an element of the class. The definition of I is consistent, for if 0 < x E E and x < a(b), Z scalar a 3 a(b) < ax:7 for suppose such an a exists. x < a(b) = x A (b - a(b)) = 0 = ax A'b = ax A (a(b) + b - a(b)),. in E = ax A a(b) + ax A (b — a(b)), in E ax A a(b) = a(b) ax A'b = a(b). But ax A'b 6 E = a(b) exists in E‘, which is not the case. Partially order the class 3 by inclusion. Each chain has an upper bound. By Zorn's lemma the class 3 has a maximal element M. By Theorem 1.5, M is prime in E.a(b) EM = b-a(b) EM. Let P=MnE. P is prime in E. We will show that 3 y E E,. y > b — a(b) such that y €.M. This proves that P is not prime in E. Let y E E, y > (b - a(b) ) b 2_y A‘b > b - a(b) #’ y A'b #’ b — a(b), since y A b E E and b — a(b) E E - E This says: b-(b-aibl) > b - y A'b i.e., a(b) > b - y A'b E E . By construction, b — y A b 6 M if yEM, then yAb EM 17 = (b - y A b) + y A b = b EEM which cannot be. So, 3 y > (b - a(b)), y E B such that y E.M. Q.E.D. Example 1.13: Let L be the set of all real bounded functions f on the point set X = (1,2,..., 00} such that f(x) #'f(m), for at most t finitely many x. This is a Riesz space with the usual. point wise operations. This has P-P'P-, 'but does not have P-P- and is not Dedekind G-complete. Let A be an arbitrary principal band in L, generated by V E L+, and denote the set (x: f(x) #'0 for at least one f 6 A} = {x:v(x)> 0} by X If v(w) = 0, then X 1' 1 consists only of a finite number of points and X2 = X - Xl contains the point m. If v(w) > 0, then X1 consists of all but a finite number of points, and X1 contains the point w.. Hence given f 6 L, we have in both cases f = f + f - xx 'with f - XX 6 A: f - xx 6 A1 X X1 2 1 2 .2 L = A @ Al , .2 L has P-P-P- L is not Dedekind complete by Theorem 0.23. To show that L does not have P9P: , consider the band A in L, defined by A (f E L f(x) = O for x = 1.3.5.'°° } then A1 [f e L o for x = 2,4,6,--- } f(x) A o Al = {f e L : f(w) = o] #'L. min 18 In this space, a prime ideal is the class of all functions vanishing at a particular point. The class of functions vanishing at a given point x0 is clearly an ideal. It is a prime ideal, since if f A g = 0, f(xo) = 0 or g(xo) = 0. These are all the prime ideals, for suppose an ideal I contains only functions which vanish on at least two points. Let the two points be x1 and x2, x1 < x2, ‘where x can 2 L4 be = m. Consider the functions h = 1 at x 3.x 1 l = 0 at x > xl h = O at x.g x1 = 1 at x > x1 hlAh2=O, but hlgl, hZKI. Suppose an ideal I contains a function k which does not vanish at any point. Let k(w) = a. = k(x) = a for all but finitely many x. Let the points y with k(y) #’a be yl, y2,..., ym. Let (IkI (yl), Ikl (yz),..., Iklym) =b>0. Let Ial Ab=C>O = k’ €I where k’(x) =c for x=l,2,..., co = I = I, J. I is not proper. Let the prime ideals be P1, P2,---, Pm. PCD has the property that its elements are functions vanishing A at all but a finite number of points. L is the class of A all bounded functions on the point set X. Clearly POD is l9 . . A A not prime in L,. for POD contains only functions which vanish at m and all but finitely many points. Example 1.14: We have noted (Theorem 0.23) that Dedekind O~¢omp1eteness is a prOperty stronger than P-P-P. The example below shows that even this property is not strong enough to imply the prnneness of P, for every prime P in E. Let L be the collection of all real bounded functions f on [0 l]such that f(x) #'f(0) for at most countably many x. This is a vector lattice with the usual pointwise operations. Given that 0 g_un < v; sup un exists in L and is the pointwise limit. Hence L is Dedekind 0— complete. To ShOW’that L does not have 'the 9-!» , let A be the band in L consisting of all f E L vanishing on [0 %J. Then A‘L is the band of all f e L vanishing on é, 1]. Any f e A satisfies f(x) #’0 for at most countably many x. The same is true for any f 6 Al and hence for any f in A (+3 Al. L 9! A (+3 Al . L does not have P-P. I0 is a prime ideal in L where I0 is the collection of all functions vanishing at 0. The Dedekind completion L of 7L is the class of all bounded functions on [0 l]. A A I0 is not prime in L,. for let A C [0 1], such that A A and AC are uncountable. Let f, g, E L such that 20 f=1 at A: g=0 at A = 0 at AC ; = 1 at AC A A o A 0 ng=0, but fEIO, gEIO, Since IO isthe collection of functions vanishing at 0 and all but countably many x in [0 l]. Henceathe example. Section 2 We had noted in Section 1 that Masterson has proved that if E has P'P- P is prime in E, if P is prime in E. (Theorem 2.1, Masterson 5) We obtain a partial converse to this result (Theorem 2.6) Masterson has mentioned the property (which we will call the property *), viz. V x E 94', the existence of y E E+ .9 a3] g.x g'y for some scalar a . He notes that if E has this property, the structure spaces of E and E are homeomorphic. He also obtains a characterization of this property (Masterson 5, Theorem 2.6). We obtain other characterizations of this property (Theorem 2.5). We show that this property is strictly stronger than P-P- (Theorem 2.6, Theorem 2.7, Example 3.5). We also Obtain a characterization of P-P° in terms of a property analogous to prOperty1r (Theorem 2.7). We also mention a property which is strictly stronger than prOperty* (Theorem 2.9, Example 2.10). Lemma 2.1: Let E be an Archimedean vector lattice and A a band in E. Then any positive element x of E can be written as x = x + x where x = sup (x ) 1 l a a 2 I A — .L . Xa E A and x2 — sap (XB)' xB E A , the sups taken in E 21 22 Proof: If x e A (+3 A‘L , there is nothing to prove. Assume x 6 E '\ A @ A‘L A ll . A l . A . A E = A (in E) (9 A (in E) (Since E has P°P~ by A Theorem 0.22) x = x1 + x2 , where x1 6 A1"L (in E) l . A x2 6 A (in E) x2=sgp(xB), XBEE VB. XBEALHnE) VB A for if xB E A for some (3 , x2 EA‘L (in E) . Also if x €E-—A<+>Al (inE)theanAy>O forsome yEA B A . J- - _ = x2£A (in E). Also xl—sgp (xa), Xa 6E V OL XO 6 A V a , for if XO 6 A‘L (in E) for some a then 0 11 . A l . . xlflA (inE). xafl E-AOA (inE) V a for if xa ’EE-A®A1(inE).then xa ’Az>0 forsome O O zEA‘L(inE) (forif xa’Az=O V ZEAl(inE), 0 then Xa ’ E A‘Ll (in E) = A, since E is Archimedean) 0 ii A = x1 E A (in E) . Q.E.D. Lemma 2.2: If E = A (43 AL , where A is an . . A A A1. A ideal in E, then E=A®A , where E, A, A are the Dedekind completions of E, A, A1 respectively, as vector lattices. A Proof: Let x E (E )+ and let x = sup xa . where " —" "' 0L 23 W-LoO-G- let, i xyg y E E+, for any element A of E is majorized by an element of E. Let y = y1 + y2 where y1 E A,, y2 6 AL. Let xa = x + Xoc where xa E A , 0‘1 2 1 l x E A V a 0‘2 x = sup x = sup (x + x ) a a a a1 a2 ‘ l = sup (x ) + sup (x ) a 1l a 0‘2 I Since x l x V d,B 0‘1 B2 A (all the sups taken in E) . Also XO 3 yl, V d , 1 Xa S-Y2 Va 2 sup (x ) g_y : SUP (X l g Y a a1 1 a d2 2 A A .L = sup (x ) E A ; sup (x ) E A a 0‘1 a OL2 A A A Corollapy 2.3: If E has P-P-, E = B O B any band B of E. Lemma 2.4 Let E be any vector lattice. Let a 6.E+ . Let B = (b 6 3+ : 3 scalar o 9 a < a (b1 A b2 A ... A bk) V integers k and all choices of b ‘b ---- b H scalar B .9 (b1 A'b A ... A bn) ll 20 I RI 2 < B a, V integers n and all choices of b1,b2,---bn.} 24 Then there exists a prime ideal P in E ,9 a E P, Note: B is the class of all elements in E+ such that if b is any finite infimum of elements of 0 B , the principal ideal generated by a is a proper subset of the principal ideal generated by bO . Also B is closed under finite infima. Prggf: Consider the class 3=[I:IanidealinE, aEI,InB=¢} 3 7! Q, for Ia E 8 , where Ia is the principal ideal generated by a. Partially order 3 by inclusion. Each chain has an upper bound. 3 has a maximal element M by Zorn's lemma. We shall prove that M is prime. Supposenot.Thenax,y€E, xAy=0 XEM,y£M. But ' ' ’ 3 ml 6M, b E B and pOSitive scalars C1, C1 3 1 b1 3C1 x-i-Cl mltalso 3 m2 EM, b2 EB and positive I I scalars C2, C2 3 b2 3C2 y+C2 m2 1:1Ah2 g(Cll’mx+C122)A(Cy+C2'm) “(CXACém2)+(CimlAczy)+(C1’mlAC2'm2) (Since x A y = 0) CI . . g sz2 + Cim1+ Clml E M . This is a Contradiction, since B is closed under finite infima and M is an ideal. 25 Theorem 2.5: The following are equivalent: ' A r A (1) Every ideal in E is of the form I, for some ideal I in E. + (2) For each u E 3 V E E and scalar (As tn> c1 ) ov'g u v . A (3) Every prime ideal in E is of the form P for some prime ideal P in E. Proof: Masterson (5, TheOrem 2.6) has proved ‘ the equivalence of statements (1) and (2). .2 We have only to prove the equivalence of (2) and (3). A (2) =q(;): Let Q be prime in E . Let A A Q 0 E = P to prove P = Q . P C Q., by definition. A A We have only to prove that Q C»P . Let u E Q n E+ by (2) 3 V E E and positive scalar a 9 av g_u g v A = av E P = VVE P = u E P (3) = (2): Assume (3) and suppose (2) is not true. Then, 3 u E E+- 9 for no x E E+ is x > u., 3 positive scalar d 9 OX S.u S.x.. Let X = [x E E+: u g x}. The hypothesis of Lemma 2.4 are satisfied 3 prime Q in E 9 u E Q and Q 0 X = Q’:then Q is not of the form P for any prime P in E, for H v E E Q Q -) V > u . This contradicts the hypothesis“ Q.E.D. 26 Theorem 2.6: Let E be an Archimedean vector A lattice and E its Dedekind completion. Let P be A A any prime ideal in E. Let P = [y 6 E : (yl g Ix}, A A for some x in P}. If P is prime in E for all A prime P in E, and every prime in E is of the form A P for some prime P in E, then E has P°P° nggf: Suppose E does not have P-P° Then 3 . l 1 band B 3 E\B®B #0“. Let E\ Bee ={Xa}aEA' A_ 11 A J- A Let xa E {Xa}d€A' E = B (in E) O B (in E) (Theorem 0.23) 0 Let Xao = X01 + x02 where x01 6 B 13y Lemma 2.1 x01 is the sup of a subset of elements of B. A A H (in E).x EB‘L (in E). 02 By theorem 2.5 3 yl €.E :) GYi g X01 g yl- Y1 6 B: for, if not y1 = Xa for some a or y1 = yll + le where y11 €.B, yl2 6 Bl (in E). y1 = xa e E \\ B e B1 (in E) i . => 3 xa: EB (in E) 9; xa Axaz > 0. xa Axa: E Iyl and 1 . . . ll . A X A x z E B (in E) which cannot be Since I C B (in E) . a a y1 Similarly yl cannot be of the form yll + yl2 where l . . 6'8, 6 B (in E) .. We have x01 < y1 where yii Yiz . . .l. . 3 y1 E B. Similarly 3 y2 E B (In E) , X02 < Y2 — L . x — x + x02 < yl + y2 6.8 O B (in E) = x 6 B O B d0 01 (in E) since B e Bl is an ideal- Contradiction! 27 Theorem 2.7: Let E be an ArchimEdean vector A lattice and E its Dedekind completion. Then E has + A+ P'P° a V a E E 3 d E E l principal bands generated by a identical (=) Let c [c Proof: B be the band generated by d > c V c 6 C . Such a A+ x 6 E 3 y 6 E ) y > x . E = d d 1 A .2 B (in E) C B a d d 2' 1 A (in E): also Let = + 6.B, d1 0 A o A 0 Bd (in E) C Ba (in E) .. l E does not have P Boslso. Suppose E 11 Let 0 \ A J. A (in E) O B (in E) . A (in E), x X0 11 6 B 02 By hypothesis 3 y1 6 E A A (in E) B (in E) Y1 y1 6 B . Similarly A A (in E) (in E) By = y2 2 X01 B e B1 .. XO— Since is an ideal. ) a < d1 and d in exists, y1 6 BX 3 y2 6 E 1 x02 < yl + y2 6 B O B and the A 1 in E are 6 E+ c < a}. Let E. Let d E E 3 for, if B 6 Bl . :0 1 d2 6 B . d1 a < d a < d l A E B C Ba (in E) 0 A o A Ba (in E)— Bd1 (in E) 'P-Then 3 ‘band B in < x e E \\ B e Bi 0 x + x where 01 02 A 6 B1 (in E) . 9 x01 < y1 and A (in E) c Bil(in E) 01 {3 x02 < y2 and 6 B1 (in E) X (in E) 6 B @ Bi, 28 2.8 A property strictly stronger than Property*. Let E be an Archimedean vector lattice and A E its Dedekind completion. Consider the property: A for x, y 6 E+ such that x < y, 3 z 6 E+ such that x < z < y’. We will show that this prOperty implies prOperty* . The converse implication is not true. There are vector lattices which have this property, but which are not Dedekind complete. /\ 13"- Proof: Let x , 2x > x I. 3 z 6 E ;) 'k 6 A A E) = Iz (in E) . Hence property . xafrPd =9 f(fma)=‘flaflf(‘lm 34 Hence f(mg) is Open in the relative topology. On the /\ other hand, if b 6 E f‘lmb 0 £010) uma : a e E, |a| < Ibl] . since bfgefl lalglbl, a6E, aEP. f is one—one onto from 5m to me) and bicontinuous, SO. f is a homeomorphism. Q.E.D. Theorem 3.3 The following are equivalent: A+ + (1) For x6E, 3 y6E f-aygxgy, A A (2) Every ideal in E is of the form I for some ideal I in E. /\ A (3) Every prime ideal in E is of the form P for some prime ideal P in E. A (4) Every prime ideal in E is of the form P for some prime ideal P in E. A .~ (5) Every ideal in E is of the form I for some ideal I in E. Proof: We have already proved the equivalence of statements (1), (2), (3) (Theorem 2.5). So it is enough to prove the equivalence of (l), (4) and (5). A (l), = ,(4): Let Q be a prfimé ideal in E. A Let P = Q n E . Let a 6 Q n (E+\ E) . By hypothesis H b 6 E+ and scalar a such that at) g_a g_b a E Q = at) E P = 'b 6 P = A C P 35 I + where A={a 6E:a’Q=P. A (4) => (1): Suppose 3 a 6E+ such that ,7! b 6 E+ and scalar (1 such that ab 3 a _<_b .... 9 Then lbl, b2, , bk . a(bl Ab2 A Abk) gag (bl Ab2 A Abk) for any finite collection of elements of E. By Theorem 1.5, there exists a prime Q in E 1) a ZQ and BCQ where B=[b6E+: b (5): Let J be an ideal in E. J n E = I is an ideal in E . Let x 6 J n El'. By hypothesis 3 y6E+ ) aygxgy.ay_<_x =y61 = BX C I where BX = [z 6 E : lZ|.S x} = J = I (5) = (1): Assume (5). Suppose (l) is not A true. 3 x 6 E+ such that there does not exist y 6 E+ such that Qty 3 x _<_ y. Let A = {y 6 E : Iyl _<_ x}. Consider the ideal generated by A A in E. Let it be IA. Consider IA: this is an ideal in A A /\ ~ E; x E IA => IA is not of the form I , for any I in E . Theorem 3.4: E has P-P e V x 6 E+ , A + a yl,y26E ylly2 and x_<_y1+y2, xcanbe 36 expressed as = x1 + x2 where xl < yl, x2 < y2 , x1, x2 6 E. Proof: ((=) Suppose E does not have P.P. = + 3 band B in E and a 6 E 9 aa 218 ® Bl /\ A /\ 13:13“L (in E) 6913JL (in E) ll . A . A a — a1 + a2, where al 6 B (in E) ,a2 6 B (in E) . . _ I I I . I By hypotheSIS, a — a1 + a2 where al g a1, a2.g a2, I I I_ I_ . al 6 E, a2 6 E. But al — al and a2 - a2 for If I I _ I I _ al < a1 or a2 < a2 . a — a1 + a2 < a1 + a2 — a s a" Contradiction. l al 6 B and a2 6 B E has P°P- (e) Let a6E+ b b 6E\+ b +b b b ' 1' 2 ' as1 2' 1" 2 Let B = {b 6 E+ : bi < b1]. Let T be the band generated ‘ = $1 : SB, YB‘L. by B in E. E T 6 . a a1 + a2, a1 6 a2 6 A /\ Al A Al Also E = T