A non-coordinatizable sectionally complemented modular lattice with a large Jonsson four-frame

A NON-COOR DINA TIZABLE SECTIONALL Y COMPLEMENTED MODULAR LA TTICE WITH A LAR GE J ´ ONSSON F OUR-FRAM E FRIEDRICH WEHR UNG Abstract. A sectionally complemente d modular lattice L i s c o or dinatizable if it is isomorphic to the lattice L ( R ) of all principal r ight ideals of a von Neumann regular (not necessarily u nital) ring R . W e say th at L has a lar ge 4 - fr ame if it has a homogeneous sequence ( a 0 , a 1 , a 2 , a 3 ) such that the neutral ideal generated by a 0 is L . J´ onsson prov ed in 1962 that i f L has a coun table cofinal sequence and a l arge 4-fr ame, then i t is coordinatizable; whether the cofinal sequence assumption could b e disp ensed with w as lef t op en. W e solve this problem by finding a non-coordinatizable sectionally complemented mo dular lattice L with a large 4-frame; it ha s cardinalit y ℵ 1 . F urthermore, L is an ideal in a complemente d m o dular lattice L ′ with a spanning 5-frame (in particular, L ′ is coordinatizable). Our pro of uses Banaschewski functions . A Banasch ewski f unction on a bounded lattice L is an antitone self-map of L that picks a complemen t for eac h element of L . In an earlier pap er, we prov ed that eve r y c ountable com- plemen ted mo dular lattice has a Banasche wski function. W e pro ve that there exists a unit-regular ri ng R of cardinality ℵ 1 and index of nilp otence 3 such that L ( R ) has no Banasc hewski function. 1. Introduction 1.1. History of the problem. The set L ( R ) of all pr incipa l rig h t ideals of a (not necessarily unital) v on Neumann regular r ing R , order ed b y inclusion, is a sublattice of the lattice o f all ideals of L ; hence it satisfies the mo dular law , X ⊇ Z = ⇒ X ∩ ( Y + Z ) = ( X ∩ Y ) + Z . (Here + denotes the addition of ideals.) Moreov er, L ( R ) is se ctional ly c omple- mente d , that it, for all principal r ight ideals X and Y such that X ⊆ Y , there exists a pr incipal r ig ht ide a l Z such that X ⊕ Z = Y . A lattice is c o or dinatizable if it is iso mo rphic to L ( R ) for some von Neumann reg ula r r ing R . In pa rticular, every co ordinatizable la ttice is sectionally complemented mo dular. (F or precis e defini- tions we refer the rea der to Section 2.) In his mono graph [25], John von Neumann prov ed the following r e sult: V on Neumann’s Co ordinatization Theorem. Every c omplemente d mo dular lattic e that admits a sp anning n -fr ame, with n ≥ 4 , is c o or dinatizable. It is not har d to find non- co ordinatizable complemented mo dular lattices. The easiest one to desc rib e is the la ttice M 7 of length tw o with seven a toms. Although Date : Septem b er 25, 2018. 2000 M athematics Subject Classific ation. 06C20, 06C05, 03C20, 16E50. Key wor ds and phr ases. lattice; complemen ted; sectionally complemen ted; mo dular; coor- dinatizable; f rame; entire; neutral; ideal; Banaschewski function; Banasc hewski measure; r i ng; v on Neumann r egular; idempotent ; larder; Condensate Lifting Lemma. 1 2 F. WE HR UNG von Neumann’s orig inal pro o f is very lo ng and techn ical (ab out 150 pages), its basic idea is fairly simple: namely , a ssume a sufficiently rich lattice- theoretical version o f a co or dinate system (the s panning n -frame, r ich ness b eing mea s ured by the conditio n n ≥ 4) to ca rry ov er the idea s in pro jective geometry underlying the construction of “von Sta udt’s algebra of throws” that makes it p os sible to go from synthetic ge ometry (geometr y descr ibed by incidence a xioms on “flats”) to analytic ge ometry (prov e statemen ts of geometry b y using c o ordinates and algebra ), see [12, Section IV.5 ]. Instead of constructing (a matr ix r ing over) a field, von Neumann’s metho d yields a reg ular ring. A p owerful generaliz a tion of von Neumann’s Co or dinatization Theor em was ob- tained by Bjarni J ´ onsson in 1960 , see [19]: J´ onsson’s Co ordinatization Theorem. Every c omplemente d mo dular lattic e L that admits a lar ge n -fr ame, with n ≥ 4 ( or n ≥ 3 if L is Arg uesian) , is c o or dina- tizable. There hav e b een many simplifications, mainly due to I. Halpe rin [13, 1 4, 15], of the pro of of von Neumann’s Co ordina tization Theor em. A s ubstantial s impli- fication of the pro of of J´ onsson’s Co or dinatization Theorem has b een achiev ed b y Christian Herrmann [16]— assuming the b asic Co or dinatization The or em for Pr o- je ctive Ge ometries , and thus reducing most of the complicated lattice c a lculations of both von Neumann’s pro o f a nd J´ onsson’s pro of to linear algebra . Now the Co or- dinatization T heo rem for Pro jective Geometries is traditionally credited to Hilb ert and to V eblen and Y oung, how ever, it is unclea r whether a c o mplete pro o f was pub- lished b efor e von Neumann’s breakthr ough in [2 5]. A very in ter esting discussion of the matter can be found in Israel Halper in’s re view o f J´ o ns son’s pap er [19], cf. MR 012017 5 (22 #1 0932). On the other hand, there is in some sense no “Ultimate Co or dinatization The- orem” for complemented mo dular lattices, as the author pr ov ed tha t there is no first-order a x iomatization for the class of all co or dinatizable lattices with unit [27]. While V on Neumann’s sufficient co ndition for coo rdinatizability requires the lat- tice hav e a unit (a spanning n - frame joins , by definition, to the unit of the lattice), J´ onsson’s sufficient condition leav es more ro o m for improvemen t. While J ´ o nsson assumes a unit in his a bove-cited C o ordinatization Theor em, a larg e n -frame do es not imply the exis tence of a unit. And indeed, J´ onsson published in 1962 a new Co or dinatization Theo rem [2 0], assuming a la rge n -fra me where n ≥ 4, where the lattice L is no longer a ssumed to have a unit (it is still sectionally complemented). . . but w her e the conclus io n is weak ened to L b eing isomorphic to the lattice of all finitely gener ated submo dules of some lo c al ly pr oje ctive mo dule ov er a regula r ring. He als o proved that if L is countable, or, more g enerally , has a co un table cofinal sequence, then, still under the exis tence o f a large n -frame, it is co or dinatizable. The ques tio n whether full co ordinatizability could b e r eached in general was left op en. In the present pap er we so lve the latter pro ble m, in the negative. Our coun- terexample is a non- co ordinatizable sectionally co mplemented mo dular lattice L , of cardinality ℵ 1 , with a large 4-frame. F urthermo re, L is isomorphic to an ideal in a complemented mo dular lattice L ′ with a s panning 5 - frame (in particular, L ′ is co ordinatizable ). Although o ur co unt erexample is constr ucted explicitly , o ur r oad to it is quite indirect. It star ts with a discov er y ma de in 1957 , by Bernhard Banaschewski [1], A NON-COORDINA TIZABLE LA TTICE 3 that on every vector space V , ov er an a r bitrary division ring, there exists an or der- r eversing (we say antitone ) map that se nds a n y subspac e X of V to a complement of X in V . Such a function w as then used in orde r to find a simple pro o f of Hahn’s Embedding T heo rem that states that every totally order ed ab elia n gr o up embeds int o a g eneralized lexico g raphic p ow er of the reals. 1.2. Banasc hewski functions on l attices and rings. By ana logy with Bana- schewski’s r esult, we define a Banaschewski fu n ction on a b ounded lattice L a s an a n titone self-ma p o f L that picks a c o mplement for each element o f L (Defini- tion 3.1). Hence Banaschewski’s ab ove-men tioned r esult from [1] states that the subspace lattice of every vector space has a Banas chewski function. This result is extended to all ge ometric (not neces s arily mo dular ) lattices in Saar im¨ aki a nd Sorjonen [26]. W e pr ov ed in [28, Theorem 4 .1 ] that Every c ount able c omplemente d mo dular lattic e has a Banaschewski function . In the pr esent pap er, we construct in Pr op o- sition 4 .4 a unital reg ular ring S F such that L ( S F ) has no Banaschewski function. The ring S F has the optimal c ardinality ℵ 1 . F urthermo re, S F has index 3 (Prop o- sition 4.5); in pa r ticular, it is unit-r egular. The construction of the r ing S F inv olves a par ameter F , whic h is any c ount - able field, a nd S F is a “ F -algebr a with quas i-inv ersio n defined b y genera tors a nd relations” in a ny lar ge enoug h v arie t y . Rela ted structur es have b een c o nsidered in Go o dearl, Menal, and Monc a si [11] and in Herrmann and Semenov a [17]. 1.3. F rom non-existence of Banasc hewski functions to fail ure of co ordina- tizability . As we are aiming to a count erexample to the a bove-men tioned problem on co ordina tization, we prov e in Theorem 6.4 a stronger negative result, na mely the non-existence o f any “ Banaschewski measure” on a certain incr easing ω 1 -sequence of elements in L . A mo dification of this ex ample, based on the 5 × 5 matrix ring ov er S F , yields (Lemma 7.4) an ω 1 -increasing chain ~ A = ( A ξ | ξ < ω 1 ) of countable sectionally complemented mo dular lattices, a ll with the same la rge 4 - frame, that canno t b e lifted, with resp ect to the L functor, by any ω 1 -chain o f reg ula r r ings (Lemma 7 .4). Our final conclusion follows from a use o f a g eneral catego rical result, called the Condensate Lifting L emma (CLL), intro duced in a pap er b y Pierre Gillibert a nd the author [9], desig ne d to relate liftings of diagr ams and liftings of obje cts . Here, CLL will turn the diagr am c ounter example of Lemma 7.4 to the obje ct c ounter ex ample of Theorem 7.5. This counterexample is a so-ca lle d c ondensate of the diagr am ~ A by a suitable “ ω 1 -scaled Bo o lean algebr a ”. It ha s cardinality ℵ 1 (cf. Theorem 7 .5). F urthermo re, it is isomor phic to an ideal o f a complemented mo dular lattice L ′ with a spanning 5 -frame (so L ′ is co or dinatizable). 2. Basic concepts 2.1. P artially o rdered sets and lattices. Let P b e a partially ordere d set. W e denote b y 0 P (resp ectively , 1 P ) the least ele men t (resp ectively , larges t element) of P when they exist, also called zer o (resp ectively , unit ) of P , and we simply write 0 (resp ectively , 1) in case P is understo o d. F ur thermore, we set P − := P \ { 0 P } . W e 4 F. WE HR UNG set U ↓ X := { u ∈ U | ( ∃ x ∈ X )( u ≤ x ) } , U ↑ X := { u ∈ U | ( ∃ x ∈ X )( u ≥ x ) } , for any subsets U and X of P , a nd we set U ↓ x := U ↓ { x } , U ↑ x := U ↑ { x } , fo r any x ∈ P . W e say that U is a lower subset o f P if U = P ↓ U . W e say that P is upwar d dir e cte d if every pair of elements of P is contained in P ↓ x for some x ∈ P . W e say tha t U is c ofinal in P if P ↓ U = P . W e define p U the least element of U ↑ p if it exists, and we define p U dually , for ea ch p ∈ P . An ide al of P is a nonempty , up ward directed, low er subset of P . W e set P [2] := { ( x, y ) ∈ P × P | x ≤ y } . F or subsets X and Y of P , let X < Y hold if x < y ho lds for all ( x, y ) ∈ X × Y . W e sha ll also write X < p (resp ectively , p < X ) instead o f X < { p } (res pectively , { p } < X ), for each p ∈ P . F or par tially ordered sets P a nd Q , a map f : P → Q is isotone ( antitone , strictly isotone , resp ectively) if x < y implies that f ( x ) ≤ f ( y ) ( f ( y ) ≤ f ( x ), f ( x ) < f ( y ), resp ectively), for all x, y ∈ P . W e refer to Birkho ff [2 ] or Gr¨ atzer [12] for basic notions of lattice theory . W e recall her e a sa mple of needed notation, terminolog y , and r esults. In any lattice L with zer o, a family ( a i | i ∈ I ) is indep endent if the equality _ ( a i | i ∈ X ) ∧ _ ( a i | i ∈ Y ) = _ ( a i | i ∈ X ∩ Y ) holds for all finite subsets X and Y of I . In ca se L is mo dular and I = { 0 , 1 , . . . , n − 1 } for a p ositive integer n , this amounts to v er ifying that a k ∧ W i ξ a nd η ∈ Λ ↓ X , 1 b , if η > ξ and η / ∈ Λ ↓ X , η X , if η < ξ a nd η ∈ Λ ↑ X , 0 b , if η < ξ and η / ∈ Λ ↑ X (w e refer the reader to Sec tio n 2.1 for the notations η X , η X ). Evide ntly , f is isotone. In particular , L V ( f ) is an endomorphism o f L V (Λ). F ro m f ↾ X = id X and the as s umption that X is a supp or t of I it follows that L V ( f )( I ) = I . On the other hand, as Λ \ { ξ } is a s uppo rt of I and f (Λ \ { ξ } ) is contained in ( X \ { ξ } ) ∪ { 0 b , 1 b } , X \ { ξ } is a suppo rt of L V ( f )( I ) (as in the pro of of Lemma 4.2). The conclusion follows.  As every element of L V (Λ) has a finite s uppo rt, we obtain immediately the following. Corollary 6 . 3. L et Λ b e a chain. Then every element I ∈ L V (Λ) has a sm al lest ( for c ontainment ) supp ort, that we shal l denote by supp I and c al l the suppo r t of I . F u rthermor e, supp I is fin ite. W e can now prov e the main result of this se ction. The F -alg ebra with quasi- inv ersion R F is defined in Sectio n 4 (cf. Figure 1). 14 F. WE HR UNG Theorem 6.4. L et F b e a c ountable field and let V b e a variety of F -algebr as with quasi-inversion c ontaining R F as an element. Then ther e exists no L V ( ω 1 ) -value d Banaschewski me asur e on the subset X F := { ˜ ξ · R V ( ω 1 ) | ξ < ω 1 } . Pr o of. The structure T := R V ( ω 1 ) is a re gular F -algebra with qua si-inv ers ion. Let t b e a term of Σ F with arity n , let Λ b e a chain, and let X = { ξ 1 , . . . , ξ n } with all ξ i ∈ Λ a nd ξ 1 < · · · < ξ n . W e sha ll write t [ X ] := t ( ˜ ξ 1 , . . . , ˜ ξ n ) ev aluated in R V (Λ) . Similarly , if n = k + l , X = { ξ 1 , . . . , ξ k } with ξ 1 < · · · < ξ k , and Y = { η 1 , . . . , η l } with η 1 < · · · < η l , we shall wr ite t [ X ; Y ] := t ( ˜ ξ 1 , . . . , ˜ ξ k , ˜ η 1 , . . . , ˜ η l ) ev aluated in R V (Λ) . If Y = { η 1 , . . . , η n } with η 1 < · · · < η n and a ∈ R V (Λ), we shall write t [ a ; Y ] := t ( a, ˜ η 1 , . . . , ˜ η n ) ev aluated in R V (Λ) . Now let ⊖ b e an L V ( ω 1 )-v alued Ba naschewski measure on X . F or a ll α ≤ β < ω 1 , there a re a finite subset S α,β of ω 1 and a term t α,β of Σ F such that ˜ β · T ⊖ ˜ α · T = t α,β [ S α,β ] · T . (6.3) As x · T = ( xx ′ ) · T for each x ∈ T , w e may as sume that the term t α,β is str ongly idemp otent . By Lemma 2.2, for ea ch α < ω 1 , there ar e an uncountable subse t W α and a finite subset Z α of ω 1 such that, setting S ′ α,β := S α,β \ Z α , S α,β ∩ S α,γ = Z α and Z α < S ′ α,β < S ′ α,γ , for all β < γ in W α . (6.4) As the simila rity type Σ F is coun table, w e may refine further the uncountable subset W α in such a way that t α,β = t α = co nstant, for all β ∈ W α . Now let α ≤ β < ω 1 . Pick γ , δ ∈ W α such that β < γ < δ . W e compute ˜ β · T ⊖ ˜ α · T = ˜ β · T ∩ (˜ γ · T ⊖ ˜ α · T ) (b y the second part of Lemma 5 .2) = ˜ β · T ∩ t α [ S α,γ ] · T , so, by using Lemma 6.1, ˜ β · T ⊖ ˜ α · T = m ( ˜ β , t α [ S α,γ ]) · T . (6.5) In particular, the supp ort of ˜ β · T ⊖ ˜ α · T (cf. Cor ollary 6.3) is cont ained in S α,γ ∪ { β } . Similarly , this supp ort is co n tained in S α,δ ∪ { β } , and so , by (6.4), supp( ˜ β · T ⊖ ˜ α · T ) ⊆ Z α ∪ { β } . (6.6) Now set k α := card Z α , for each α < ω 1 , and define a new term u α by u α ( x , y 1 , . . . , y k α ) := m  x , t α ( y 1 , . . . , y k α , 1 , . . . , 1)  , (6.7) where the n umber of o ccurrences of the consta n t 1 in the right hand side of (6.7) is equal to a rity( t α ) − k α . As m is s trongly ide mp otent, so is u α . Claim 1. The e quality ˜ β · T ⊖ ˜ α · T = u α [ ˜ β ; Z α ] · T holds for al l α ≤ β < ω 1 such that Z α ⊆ β + 1 . A NON-COORDINA TIZABLE LA TTICE 15 Pr o of of Claim. Pic k γ ∈ W α such that β < S ′ α,γ (b y (6.4), this is p ossible) and define the is otone map f : ω 1 → ω 1 ⊔ { 1 b } by the rule f ( ξ ) := ( ξ ( if ξ ≤ β ) 1 b (if ξ > β ) , for each ξ < ω 1 . Every element of Z α ∪ { β } lies b elow β , thus it is fixed b y f , while f sends e a ch element of S ′ α,γ to 1 b . Hence, by applying the mor phis m L V ( f ) to each side of (6.5) and by using the definition (6.7), we obtain L V ( f )( ˜ β · T ⊖ ˜ α · T ) = u α [ ˜ β ; Z α ] · T . On the other ha nd, a s every element of Z α ∪ { β } is fix ed by f , it follows from (6.6) that ˜ β · T ⊖ ˜ α · T is fixed under L V ( f ). T he conc lus ion follows.  Claim 1. As u α is a str ongly idemp otent term, the e le ment e α := u α [1; Z α ] is idemp otent in T . Claim 2. The r elation T = ˜ α · T ⊕ e α · T holds for e ach α < ω 1 . Pr o of of Claim. Let β < ω 1 with α < β and Z α < β , a nd define an isotone ma p g : ω 1 → ω 1 ⊔ { 1 b } by the rule g ( ξ ) := ( ξ (if ξ < β ) 1 b (if ξ ≥ β ) , for each ξ < ω 1 . F ro m Claim 1 it follows that ˜ β · T = ˜ α · T ⊕ u α [ ˜ β ; Z α ] · T , th us, apply ing the 0-lattice homomorphism L V ( g ), we obtain T = ˜ α · T ⊕ u α [1; Z α ] · T = ˜ α · T ⊕ e α · T .  Claim 2. Claim 3. The c ontainment e β · T ⊆ e α · T holds, for al l α ≤ β < ω 1 . Pr o of of Claim. Pic k γ < ω 1 such that β < γ and Z α ∪ Z β < γ . W e compute u β [ ˜ γ ; Z β ] · T = ˜ γ · T ⊖ ˜ β · T (b y Claim 1) ⊆ ˜ γ · T ⊖ ˜ α · T (b y the monotonicity assumption o n ⊖ ) = u α [ ˜ γ ; Z α ] · T (b y Claim 1) , th us, as u α [ ˜ γ ; Z α ] is idempo tent , u β [ ˜ γ ; Z β ] = u α [ ˜ γ ; Z α ] · u β [ ˜ γ ; Z β ] . (6.8) Now define an iso tone map h : ω 1 → ω 1 ⊔ { 1 b } by the rule h ( ξ ) := ( ξ (if ξ < γ ) 1 b (if ξ ≥ γ ) , for each ξ < ω 1 . By a pply ing R V ( h ) to the equation (6.8), we obtain that e β = e α · e β . The co nclusion follows.  Claim 3. By Cla ims 2 and 3, the family ( e α · T | α < ω 1 ) defines a partia l Banaschewski function on { ˜ α · T | α < ω 1 } in L V ( ω 1 ) = L ( R V ( ω 1 )). This contradicts the result of Pro po sition 4.4(ii).  16 F. WE HR UNG 7. A non-coordina tizable l a ttice with a large 4 -frame A weak er v ariant of J´ onss o n’s Pro ble m, of finding a non-co o rdinatizable sec- tionally complemented mo dular lattice with a large 4-frame, asks for a diagr am c ounter example instead of an obje ct c ounter ex ample . In order to so lve the full pro b- lem, we shall firs t settle the w eaker version, b y finding an ω 1 -indexed diag ram of 4 / 5-entire countable sectionally complemented mo dular lattices that cannot be lifted with resp ect to the L functor (cf. L e mma 7.4). The full solution of J´ ons s on’s Pr oblem will then be achiev ed by inv o king a to o l from c ate gory the ory , int ro duced in Gillib ert a nd W ehrung [9], designed to turn diagram counterexamples to ob ject counterexamples. This to ol is ca lled there the “Condensate Lifting Lemma” (CLL ). The genera l c ontext of CLL is the following. W e ar e g iven c ate gories A , B , S together with functors Φ : A → S and Ψ : B → S , such that fo r “many” ob jects A ∈ A , there exists a n ob ject B ∈ B suc h tha t Φ( A ) ∼ = Ψ( B ). W e ar e try ing to find an assignment Γ : A → B , “as functor ial as po ssible”, such that Φ ∼ = ΨΓ on a “lar g e” sub categ ory o f A . Roughly sp eaking, CLL sta tes that if the initial categor ical data can b e augmented by sub categ ories A † ⊆ A and B † ⊆ B (the “small ob jects”) together with S ⇒ ⊆ S (the “double arrows”) such that ( A , B , S , Φ , Ψ , A † , B † , S ⇒ ) forms a pr oje ctable lar der , then this can be done. Checking la rderho o d, although somehow tedious, is a relatively easy matter, the least trivia l p oint, alrea dy chec ked in [9], being the verification of the L¨ o wenheim-Sk olem Pr op erty LS r ℵ 1 ( B ) (cf. the pro o f of Lemma 7.2). Besides an infinite combinatorial lemma by Gillibert, namely [8, Pro po sition 4 .6], we shall need only a small par t of [9]; basic ally , refer ring to the num b er ing used in version 1 of [9] (which is the current version a s to the present writing), — The definition of a pro jectabilit y witness (Definition 1 -5.1 in [9]). — The definitio n of a pro jectable lar der (Definit ion 3-4.1 in [9]). Stro ng larders will not b e used. — The statemen t of CLL (Lemma 3-4 .2 in [9]), for λ = µ = ℵ 1 . This statement in volv es the catego ry Bo ol P (Definition 2- 2 .3 in [9 ]), here for P := ω 1 , and the definition of B ⊗ ~ A for B ∈ Bo ol P and a P -indexed diagram ~ A . These constructions are ra ther easy and only a few of their prop erties, recorded in Chapter 2 of [9], will b e used. A full understand- ing of lifters , or o f the P -sca led Bo o lean alg ebra F ( X ) in volved in the statement of CLL, is not needed. — P a rts o f Cha pter 6 in [9], that are, ess e ntially , easy categor ic al statements ab out r egular ring s . W e shall consider the similarit y type Γ := (0 , ∨ , ∧ , a 0 , a 1 , a 2 , a 3 , c 1 , c 2 , c 3 , I ), where 0, 1, the a i s, and the c i s are symbols o f constant, b oth ∨ and ∧ a re symbo ls of binary op er ations, and I is a (unary) pr edicate s y m bo l. F urthermore, we consider the ax iom system T in Γ tha t states the following: (LA T) (0 , ∨ , ∧ ) defines a s ectionally co mplemen ted mo dular lattice struc tur e; (HOM) ( a 0 , a 1 , a 2 , a 3 ) is independent and a 0 ∼ c i a i for each i ∈ { 1 , 2 , 3 } ; (ID) I is a n ideal; (REM) ev er y element of I is subp ersp ective to a 0 and disjoint from L 3 i =0 a i ; (BASE) ev er y element lies b elow x ⊕ L 3 i =0 a i for some x ∈ I . In par ticular, (the underly ing lattice o f ) ev er y mo del for T is 4 / 5-entire (cf. Definition 2.1), so it has a lar g e 4-frame. A NON-COORDINA TIZABLE LA TTICE 17 Observe that every axio m of T has the form ( ∀ ~ x )  ϕ ( ~ x ) ⇒ ( ∃ ~ y ) ψ ( ~ x , ~ y )  for finite conjunctions of atomic formulas ϕ and ψ . F or example, the axiom (RE M) can b e written ( ∀ x )  I ( x ) ⇒  x ∧ ( a 0 ∨ a 1 ∨ a 2 ∨ a 3 ) = 0 and ( ∃ y )( x ∧ y = a 0 ∧ y = 0 and x ≤ a 0 ∨ y )   . It follows that the categ ory A of all mo dels o f T , with their homomor phisms, is closed under arbitra ry pr o ducts and direct limits (i.e., dir ected colimits) of mo dels . Denote by S the ca teg ory of a ll sectionally co mplemen ted mo dular la ttices with 0-lattice homomor phisms, and deno te by Φ the forg etful functor from A to S . Denote by B the category of all von Neumann regular ring s with ring homomor- phisms, and take Ψ := L , which is indeed a functor from B to S . Denote by A † (resp ectively , B † ) the full sub catego ry of A (resp ectively , B ) co n- sisting of all c ountable structures. Denote by S ⇒ the categor y of all sectionally complemented modula r lattices with surje ctive 0- lattice homomorphisms. The morphisms in S ⇒ will b e called the double arr ows of S . Our first ca teg orical statement ab out the da ta just introduced in volves the left lar ders developed in [9 , Section 3.8]. Lemma 7.1. The quadruple ( A , S , S ⇒ , Φ) is a left lar der. Pr o of. W e recall that left larder s are defined by the following prop erties: (CLOS( A )) A has all sma ll dire cted colimits; (PROD( A )) A has all finite nonempty pro ducts; (CONT(Φ) Φ pre s erves all small direc ted colimits; (PROJ(Φ , S ⇒ )) Φ sends any extended pro jection o f A (i.e., a direct limit p = lim − → i ∈ I p i for pro jections p i : X i × Y i ։ X i in A ) to a double arrow in S . All the corresp onding verifications are straig h tforward (e.g., every extended pro- jection f is surjective, th us Φ( f ) is a double arrow).  Our second catego rical statement sta tes something ab out the more inv olved no- tion, defined in [9, Section 3.8], o f a right λ -lar der . W e shall also use the notions, int ro duced in that pap er, of pr oje ctability of right larder s. The following result is a particula r case, for λ = ℵ 1 , of Theorem 6 -2.2 in (version 1 of ) [9]. Lemma 7. 2. Denote by S † the class of al l c ountable se ctional ly c omplemente d mo dular lattic es. The n the 6 -uple ( B , B † , S , S † , S ⇒ , L ) is a pr oje ctable right ℵ 1 - lar der. Pr o of. Right lar derho o d amoun ts here to the conjunction of the tw o following s tate- men ts: • PRES ℵ 1 ( B † , L ): The lattice L ( B ) is “weakly ℵ 1 -presented” in S (which means, here, c ount able ), for each B ∈ B † . • LS r ℵ 1 ( B ) (for ev er y ob ject B o f B ): F or ev ery countable sectionally comple- men ted mo dular lattice S , every surjective lattice ho momorphism ψ : L ( B ) ։ S , and every sequence ( u n : U n ֌ B | n < ω ) of mono mor- phisms in B with a ll U n countable, ther e exists a monomor phism u : U ֌ B in B , lying a bove all u n in the subob ject ordering, s uch that U is countable and ψ ◦ L ( u ) is surjective. Both s tatement s are verified in [9, Chapter 6 ].  18 F. WE HR UNG Now bring ing together Lemmas 7 .1 and 7 .2 is a trivial matter: Corollary 7 .3. The 8 -uple ( A , B , S , A † , B † , S ⇒ , Φ , L ) is a pr oje ctable ℵ 1 -lar der. The following cruc ia l result ma kes a n essential use of our work on Banaschewski functions in Section 4. Lemma 7.4. Ther e ar e incr e asing ω 1 -chains ~ A = ( A ξ | ξ < ω 1 ) and ~ A ′ = ( A ′ ξ | ξ < ω 1 ) of c ountable mo dels in A , al l with a u nit, such t hat the fol- lowing statements hold: (i) Φ ~ A c annot b e lifte d, with r esp e ct to the L functor, by any diagr am in B . (ii) A ξ is a princip al ide al of A ′ ξ , for e ach ξ < ω 1 . (iii) All the mo dels A ′ ξ shar e t he same sp anning 5 -fr ame. Pr o of. W e fix a countable field F and we define regular F -algebra s with quas i- inv ersion by R ξ := R F ( ξ ) (as defined in the co mments just b efore Prop os ition 4.5) and S ξ := R 5 × 5 ξ , for any ordinal ξ . W e s et R := R ω 1 and S := S ω 1 , and we ident ify R ξ with its canonica l image in R , for each ξ < ω 1 (this requires P r op osi- tion 4.3). W e denote by ( e i,j | 0 ≤ i, j ≤ 4 ) the c a nonical s y stem of matr ix units of S , s o P 0 ≤ i ≤ 4 e i,i = 1 and e i,j e k,l = δ j,k e i,l (where δ denotes the K roneck er symbol) in S , for all i, j, k , l ∈ { 0 , 1 , 2 , 3 , 4 } . W e denote b y ψ := (( e i,i S | 0 ≤ i ≤ 4) , (( e i,i − e 0 ,i ) S | 1 ≤ i ≤ 4)) the c a nonical spanning 5- frame of L ( S ). F urthermo re, we se t e := P 0 ≤ i ≤ 3 e i,i , b := e 4 , 4 , and b ξ := ˜ ξ · b for each ξ < ω 1 . Observe that e , b , a nd all b ξ are idemp otent, and that 1 = e ⊕ b and b ξ E b in S . W e set U ξ := ( e + b ξ ) S , for each ξ < ω 1 , and A ′ ξ := canonical copy of L  ( R ξ +1 ) 5 × 5  in L  R 5 × 5  , A ξ := ideal of A ′ ξ generated by U ξ , for each ξ < ω 1 . In particula r, A ′ ξ is a countable complemented sublattice of L ( S ) containing ψ while A ξ contains φ := (( e i,i S | 0 ≤ i ≤ 3) , (( e i,i − e 0 ,i ) S | 1 ≤ i ≤ 3)), the cano nical spanning 4 -frame of the principa l ideal L ( S ) ↓ eS . In ea ch A ξ , we interpret the co nstant a i by e i,i S , for 0 ≤ i ≤ 3, and the co n- stant c i by ( e i,i − e 0 ,i ) S , for 1 ≤ i ≤ 3. F ur ther more, we in ter pret the predicate symbol I of Γ in each A ′ ξ by A ′ ξ ↓ b S , and in each A ξ by A ξ ↓ b ξ S . It is straightforw a rd to verify that we thus obtain incr easing ω 1 -chains ~ A and ~ A ′ of co unt able mo de ls in A . W e claim that there is no L ( S )-v alued Bana s chewski measure on { U ξ | ξ < ω 1 } . Suppo se otherwise. As U ξ = eS ⊕ b ξ S and b ξ S ⊆ bS , with eS ⊕ bS = S in L ( S ), there exists, b y Lemma 5.3, an  L ( S ) ↓ bS  -v alued Ba naschewski measur e on { b ξ S | ξ < ω 1 } . How ever, it follows from [20, Lemma 10.2] that L ( S ) ↓ b S is isomor phic to L ( R ), via an isomorphism that sends b ξ S to ˜ ξ R , for each ξ < ω 1 . Thus there exists an L ( R )- v alued Banas chewski measure on { ˜ ξ R | ξ < ω 1 } . This contradicts Theo rem 6.4. An y lifting of ~ A , with resp ect to the functor L , in B ar ises from an ω 1 -chain B 0 ⊂ B 1 ⊂ · · · ⊂ B ξ ⊂ · · · of regular r ing s, and it can b e represented b y the commutativ e diag ram o f Figure 2, for a system ( ε ξ | ξ < ω 1 ) of isomorphisms. It follows from Lemma 2.4 that B ξ is unital, for each ξ < ω 1 . Denote by 1 ξ the unit of B ξ , and set U β ⊖ U α := ε β  (1 β − 1 α ) · B β  , for all α ≤ β < ω 1 . A NON-COORDINA TIZABLE LA TTICE 19 A 0   / / A 1   / / · · · · · ·   / / A ξ   / / · · · L ( B 0 ) ε 0 ∼ = O O   / / L ( B 1 ) ε 1 ∼ = O O   / / · · · · · ·   / / L ( B ξ ) ε ξ ∼ = O O   / / · · · Figure 2. A lifting of Φ ~ A with resp ect to L Let α ≤ β ≤ γ < ω 1 . F ro m the co mm uta tivit y of the diagr am in Figur e 2 it follows that U α = ε β (1 α · B β ). Hence, by applying the lattice isomorphism ε β to the relation B β = 1 α · B β ⊕ (1 β − 1 α ) · B β , we obtain the rela tio n U β = U α ⊕ ( U β ⊖ U α ). F urthermo re, from 1 α E 1 β E 1 γ it follows that 1 γ − 1 α = (1 γ − 1 β ) ⊕ (1 β − 1 α ) in Idemp B γ , thus (1 γ − 1 α ) · B γ = (1 γ − 1 β ) · B γ ⊕ (1 β − 1 α ) · B γ in L ( B γ ), thus, applying ε γ to each side of that relation, we obtain U γ ⊖ U α = ( U γ ⊖ U β ) ⊕ ε γ  (1 β − 1 α ) · B γ  = ( U γ ⊖ U β ) ⊕ ε β  (1 β − 1 α ) · B β  (see Figure 2) = ( U γ ⊖ U β ) ⊕ ( U β ⊖ U α ) . Therefore, ⊖ defines an L ( S )-v a lued Banaschewski measure on { U ξ | ξ < ω 1 } , which we just prov ed imp ossible.  Observe that all the A ′ ξ s share the s a me unit, while the ω 1 -sequence formed with all the units o f the A ξ s is increasing . Theorem 7.5. Ther e exists a non-c o or dinatizable, 4 / 5 -ent ir e se ctional ly c omple- mente d mo dular lattic e L of c ar dinality ℵ 1 , which is in addition isomorphic to an ide al in a c omplemente d mo dular lattic e L ′ with a s p ann ing 5 -fr ame ( so L ′ is c o or- dinatizable ) . Pr o of. W e use the notation and terminology o f Gillibert and W ehrung [9]. It fol- lows fr o m Gillib ert [8, Pro po sition 4.6] that there exists an ℵ 1 -lifter ( X, X ) of the chain ω 1 such that card X = ℵ 1 . Consider the diagrams ~ A and ~ A ′ of Lemma 7 .4, a nd observe that bo th A ξ and A ′ ξ belo ng to A † (i.e., they are co un table), for ea ch ξ < ω 1 . W e for m the c ondensates L := Φ  F ( X ) ⊗ ~ A  and L ′ := Φ  F ( X ) ⊗ ~ A ′  . F ro m card X ≤ ℵ 1 it follo ws that the ω 1 -scaled Bo olean algebra F ( X ) is the directed colimit of a direct system o f at most ℵ 1 finitely presented o b jects in the category Bo ol ω 1 . It follows that card L ≤ ℵ 1 and card L ′ ≤ ℵ 1 . W e shall prov e that L is not co ordinatizable ; in particula r , b y [20, Theorem 1 0.3], card L = ℵ 1 . Suppo se that there exists an isomorphism χ : L ( B ) → L , for s o me regular ring B . By CL L (cf. [9, Lemma 3 -4.2]) together with Coro llary 7 .3, there exists a n ω 1 - indexed diag ram ~ B in B such that L ~ B ∼ = Φ ~ A . This contradicts Lemma 7.4. There- fore, L is not co ordina tizable. F urthermo re, F ( X ) ⊗ ~ A is a direct limit of finite direct pro ducts of the for m Q n i =1 A ξ i , where the s hap e o f the indexing system depends only on X . As A ξ is an ideal o f A ′ ξ for each ξ < ω 1 , Q n i =1 A ξ i is an ideal of Q n i =1 A ′ ξ i at each of thos e places. Therefore, taking direct limits, we obtain that F ( X ) ⊗ ~ A is isomorphic to an ideal 20 F. WE HR UNG of F ( X ) ⊗ ~ A ′ , so L is a n ideal o f L ′ . As the c lass of all lattices with a s panning 5-frame is closed under finite pro ducts and dir ected co limits and as all A ′ ξ s have a spanning 5- frame, L ′ also ha s a s panning 5-fr a me.  Theorem 7.5 provides us with a non- c o ordinatizable ideal in a co or dinatizable complemented mo dular lattice of car dinality ℵ 1 . W e do not kno w whether an ideal in a c ountable co or dinatizable sectionally co mplemen ted mo dular la ttice is co ordinatizable . As the lattice L of Theo rem 7.5 is 4 / 5-entire and sectionally complemen ted, it has a larg e 4-fra me. Hence it solves neg a tively the pro blem, left op en in J´ onsson [20], whether a sectiona lly co mplemented mo dular lattice with a la rge 4- frame is co or - dinatizable. R emark 7 .6 . As the lattice L o f Theor em 7.5 has a la rge 4- frame, every principal ideal of L is co ordinatiza ble. Indeed, fix a large 4-fra me α = ( a 0 , a 1 , a 2 , a 3 , c 1 , c 2 , c 3 ) in L and put a := L 3 i =0 a i . Every principa l ideal I of L is contained in L ↓ b for some b ∈ L such tha t a ≤ b . As α is a lar ge 4 -frame of the complemented mo dula r lattice L ↓ b and by [19, Theorem 8.2], L ↓ b is co ordina tizable. As I is a principal ideal of L ↓ b , it is also co ordina tizable (cf. [20, Lemma 1 0.2]). R emark 7 .7 . It is proved in W ehrung [27] that the unio n of a chain of c o ordinatizable lattices may not be co ordinatizable . The lattices conside r ed there are 2-distributive with unit. Theor em 7.5 extends this negative result to lattices (without unit) with a la rge 4-fra me. F urthermore, it also shows tha t an idea l in a co or dinatizable lat- tice L ′ may not b e co ordinatizable, even in case L ′ has a spanning 5-frame. By contrast, it follows from [20, Lemma 10.2] that any princip al ideal of a co ordina - tizable lattice is co o rdinatizable. It is a lso o bs erved in [27, Prop os itio n 3.5] that the clas s of co ordinatizable lattices is clo s ed under homomorphic images, reduced pro ducts, and taking neutra l ideals . It is prov ed in W ehrung [27] that the class of all co or dinatizable lattices with unit is not first-or der. The lattices considered ther e are 2-distributive (thus without non-trivial homog eneous sequences) w ith unit. The following re sult extends this negative res ult to the cla ss of all lattices (without unit) admitting a la rge 4-fra me. Corollary 7.8 . The class of al l c o or dinatizable se ctional ly c omplemente d mo dular lattic es with a lar ge 4 -fr ame is not first-or der definable. Pr o of. Fix a la rge 4-frame α =  ( a 0 , a 1 , a 2 , a 3 ) , ( c 1 , c 2 , c 3 )  in the la ttice L of Theo- rem 7 .5, and put a := a 0 ⊕ a 1 ⊕ a 2 ⊕ a 3 . As L is 4 / 5-entire, it satisfies the firs t- o rder statement, with parameter s from { a 0 , a } , ( ∀ x )( ∃ y )( x ≤ a ⊕ y and y . a 0 ) . (7.1) Let K be a countable elementary sublattice of L co nt aining all the se ven entries of α . As L sa tis fies (7 .1) , so do es K , thus α is a la rge 4 -frame in K . It follows from [20, Theorem 10.3] tha t K is co ordinatiza ble. O n the o ther hand, L is not co ordinatizable a nd K is an elementary sublattice o f L .  The following definition is intro duced in [28, Definition 5.1]. Definition 7. 9. A Banaschewski tr ac e on a lattice L with ze ro is a family ( a j i | i ≤ j in Λ) o f elements in L , where Λ is a n upw ard directed partially ordered set with zero, s uch that A NON-COORDINA TIZABLE LA TTICE 21 (i) a k i = a j i ⊕ a k j for all i ≤ j ≤ k in Λ; (ii) { a i 0 | i ∈ Λ } is co final in L . W e prov ed in [28, Theorem 6.6] that A se ctional ly c omplemente d mo dular lattic e with a lar ge 4 -fr ame is c o or dinatizable iff it has a Banaschewski tr ac e . Hence we obtain the following r e sult. Corollary 7 . 10. Ther e exists a 4 / 5 -ent ir e se ctional ly c omplemente d mo dular lat- tic e of c ar dinality ℵ 1 without a Banaschew ski tr ac e. 8. Ackno wledgment I thank Luca Giudici for his many thoughtful and ins piring comments on the pap er, in particular fo r his example quo ted in Rema rk 4.7. References [1] B . Banaschewski, T otalgeordnete Mo duln (German), Arch. M ath. 7 (1957), 430–440. [2] G. 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Saarim¨ aki and P . Sorj onen, On Banasc hewski functions in l attices, Algebra U nive r salis 28 , no. 1 (199 1), 103–118. [27] F. W ehrung, V on Neumann coor dinatization is not first-order, J. Math. Log. 6 , no. 1 (2006 ), 1–24. [28] F. W ehrung, Co ordinatization of lattices b y regular rings without unit and Banaschewski functions, Algebra Univ ersal i s, to app ear. LMNO, CNRS UMR 6139, D ´ ep ar tement de M a th ´ ema tiques, BP 5186 , Universit ´ e de Caen, Campu s 2 , 14032 Caen cedex, France E-mail addr ess : wehrung@math .unicaen .fr URL : http://w ww.math. unicaen.fr/~wehrung