On lattice-dilations and contractions in f-rings

Pub . Ma . UAB N° 20 Se . 1980 Ac es VII JMHL ON LATTICE-DILATIONSANDCONTRACTIONS IN -RINGS Joan T ias Pai d Dp . de Ma emá iques i Es adís ica, E.T .S.A.B . Uni e si a Poli bcnica de Ba celona ABSTPACT . This pape spli s b oadly in o wo ela ed pa s, conce ned espec- i ely wi h gene alized idempo en s(associa ed o supe uni ies and subuni ies) andwi h la ice-dila ions and con ac ions in an - ing A . I u is a supe uuii- y, we cha ac e ize he mappings F :A-->A sa i ying IF(x)-F(y)I = uIx-y1 (u-di- la ions)as he mappings o he o m F(x)= x(u-2e) b, wi he being a gene ali .- zed idempo en , and ob ain an analogous esul o la ice-con ac ions . The se o he hcnageneous ones(bo h cases) a e p o ed obe Boolean algeb as . Ou e minology and no a ions a e mos ly s anda d, and ollowwidely L1] . Recall ha inan -e- ing A , u is a supe uni y 133 i ux Axu .>,,x holds o e- e y x >/ 0, and s is a subuni y L6 ] i 0 < sx < x , 0 < xs ; x  o e e y x> 0 . Now, i ollows a summa y o esul swi hou p oo s . 1 . Gene alized ideaqo en s . In an -¿- ing A, we in oduce he ollowing de i- ni ions : a) i u is a s u T ne uni y, hen eEA is a u- idempo en i eu= ue= e2 . b) i s is a subuni y, hen e E A is an s- idempo en i es = se = e 2 . The es- pec i e se s will be deno ed by I(u) and I(s) . We i s show ha in an - ing A , I (p) = { x E A 1 x A (p-x) = 0 } i p is a subuni y o a supe uni y, and ob ain as a consequence : Theo em 1 .I(p) is a Boolean algeb a wi h he o de ing o he ing . 2 . Boolean al  as o gene alized idempo en s . On accoun o I (p) ha ing no special p ope y whi .c h an a bi a y Booleanalgeb a needno na e, we ha eana- lized i s boolean p ope ies in eonnec ion wi h la ice and algeb aic- heo e ic p ope ies o he ing . Since I (p) is closed by aking a bi a y sup ema and in ima, i is easy o ela e he o de oomple eness p ope ies o A wi h ~le e ness p ope ies o I(p) . We pay conside able a en ion o p ojec able - ings, ha is, hose o which a l ® -L-= A holds o e e y a E A . They a e specially in e is ing in his con ex in iew o he ollowing esul o - ings wi h a supe uni y u : Theo em 2 . A is p ojec able i and only i he pola o e e y elemen is he pola o a unique u-idempo en . Fo he "only i " pa o Th .2, i su ices p o ing ha i a E A, hen al =  e l , being e he p ojec ion o u on o á" « L . Some cceple eness and p ojec ion p ope ies :L-,ply ha he - ing A be p ojec able . Fo ins ance, hose o he main inclusion heo em [4J , and o- he s . We ha e p o ed he e ha i I(u) is eon ex, hen A is p ojec able . F cm Th .2 we ob ain ha i A is p ojec able and non o ally o de ed, hen I(u) is non i ial . Sane esul s sugges he in e es o s udying he subse Pu (A) = e - ' - 1 eE I(u)} o all he pola s P(A) o he ing . In his connexion we p o e : Theo em 3 . Wi h he o de ing o P(A), Pu (A) is a Boolean subalgeb a o P(A), iscmo phic o he algeb a I(u) . Mo eo e , Pu (A) is a sub-la ice o PP(A), he la ice o all p incipal pola s, and a sualgeb a o he Boolean algeb a o di ec sum :nands . I P(A) is supe a onic, hen I(u) is ini e . I u' is ano he supe uni y, hen I(u) and I(u') a e isom phic in he o- llowing cases : a) A is Dedekind-cade e ; b) I(u) and I(u') a e comple e Boolean algeb as and a e con ex in A . The p oo o TM uses mainly he decomposi ion A = e l ® (u-e) 1 i eeI(u), and he ac ha i e l , e 2 e I(u) and el = e2  hen el =e 2 . The isom phism o he s a emen is gi en by  I(u)--> P u (A) , e~ (u-e) 1 . By using Th .3 in he p ojec able case, we ob ain : Theo em 4 . Le A be a p ojec able - ing . a) I(u) is a omic in he ollo- wing cases : 1) P(A) is a cenic ; 2) A is basic o ~le ely dis ibu i e . b) I A has a ini e basis, hen I(u) is ini e . 3 . Boolean al  as o la ice - dila ions . I u is a cen al supenini y o he - ing A, we gene alize he no ion o  ¿-isone y ( [21 , [5'1, [6 1) by conside ing he mappings F : A->A ha sa is y IF(x)-F(y)j = uIx-y1 o e e y x,yE A . They will be called la ice - dila ions o u- dila ions , since IF(x)- F(y)I> 1x-y1 . We deno e by H u (A) he 'se o all hcmogeneous u-dila ions, ha is, hose o which F(0) = 0 . On he o he hand, i e E I(u), we conside he mapping c e :A -~ A , ~ e (x)= x(u-2e) and se Z u (A) _ { ' e leeI(u)1 The undamen al esul we ha eob ained ncw ollows : Theo em 5 . a) Hu (A)= o~) u (A) . b) E e y u-dila ion F is o he o m F(x)= x(u-2e)T b, being b e A and eE I(u) . c) E e y Fe Hu (A) is a ham ecy o a- io a, wi h I a I = u Pa a)'o Th .5 has been p o ed by means o a sui able ep esen a ion o A as a subdi ec p oduc o o ally o de ed ings, and he ac ha o 'a o allyo de ed ing wi h a supe uni y u, he only u-dila ions a e O' 6 (x)= ux o e e y x, and a- u (x)= -ux o e e y x . Pa c) ollows on accoun o e E I(u) being a componen o u . 1 Now, i we conside he Boolean ing s uc u e o he Boolean algeb a I(u), hen Th .5 enables us o endow Hu (A) wi h a Boolean s uc u e : Theo em 6 . Wi h he ope a ions  (a - e ® é,) (x)= x(u-21e-e'I)  and (oy e ,Y a e' ) ( x ) =x (u-2 (e n e' )),  (H u (A),  , IK ) is a Boolean ing wi h uni y, isomo phic o he Boolean ing I(u) . Nowin iewo he isceno phism Hu(A)='I(u) and he heo ems 3 and 4, we i ha e de i ed he co esponding p ope ies o he Booleanalgeb a Hu (A), bu we shall no explici ly men ion hemhe e . ~ e , i . iswo h no ing ha i A is p ojec able and non o ally o de ed, hen he e exis non i ial u-iila ions . The u-dila ions ' o and o - u a e in e es ing since we ha e : Theo em 7 . I F E Hu (A) , hen he e exis s a unique decanposi ion A= B$C, wi h - B,C being ¿-ideals, o which FIB '7 0 and FI C Qu . Indeed, by Th .5  F= o e , o sane e e I(u), and i su ices aking B= e l and C= (u-e) l . Theo em 7 enables us o gi e sane geome ic in e p e a ion o ha ogeneous u-dila icns, especially byneanso he coneep o la iee axial sycm y . Recall can [ 6 ] ha i aeA , hen :A-s A . is a la ice axial symme y o axis a i : 1) is a g oup hamam phism : 2) A=<aj® aL and 3) La> I , 1 a1= -I , wi h I being he iden i y mapping . Then we can pa ially eph a- se TM : E e y hcuegeneous u-dila ion é is a homo ecy o a io u on o hogonal di ec ions, ollowed by a la ice axial symme y wi h espec o oneo ha di ec ions(o axis u-e), o , which is he same : i is a la ice axial symme y ollowed by a hano ecy o a io u . Con e sely, e e y la ice axial symme y, ollowed bya ha o ecy o a io u is a hanogeneous u-dila- ion . In he cou se o ou s udy he se B o all squa e oo s o u 2 has na u a- lly a isen . We ha e pM ed ha B =La¡ la ¡= u} . Mo eo e , Theo em 8 . Wi h he same o de ing o he ing, B is a Boolean algeb a, ha is isamo phic o he Booleanalgeb a I(u) . Hence isamo phic o Hu (A) . The p eceding iscmo phism is gi en by I(u) -B , e -~2e-u . I is pos- sible now o ans e o Bmanyo he p ope ies ha could be asse ed o he Booleanalgeb a I(u) . 4 . La ice - oon ac i e mappings . I s is a cen al subuni y o he - ing A, we can de ine "mu a is mu andis" he concep o la ice - con ac ion (s- con ac ion ) and hoimgeneous s-con ac ion by only in e changing u by s in he de ini ion . Wi h ce ainaddi ional assump ions on A , i is possible o de elop a heo y o s-con ac ions, ha is pa allel o ha o u-dila- ions, hough less sa is ac o y in some aspec s . The di icul yappea s when se me o he p ope ies ha a e alid o u-idemoo en s do no hold o s-idempo en s . Fo ins ance, he decomposi ich A= el e (u-e)' L is no mo e a- lid o e e ye E I(s) . I emains alid howe e i A is Dedekind-comple e o i s is a o mal uni y . O he p ope ies s ill hold in absence o nonze o nilpo en elemen s . REFERENCES ~1] Biga d,A ., Keimel,K .,Wol ens ein,S . : G oupes e anneaux é iculés . Lec u e N . in Ma h . 608 . Be lin-Heidelbe g N .Y ., 1978 . 12] G ané,J . : Sob e las isome ias de los g upos y los anillos e iculados . Pub . Uni . d e Ba celona, 1978 . 3] Hen iksen,M .,Isbell,J .R . : La ice-O de ed Rings and Func ion Rings . Paci ic Ma h . J . 12(533-565), 1962 . 4] Luxembu g,W .,Zaanen,A . : Riesz Spaces I . Ams e dam,1971 . 15] Swamy, K .L.N . : Isome ies in Au ome ized La ice O de ed G oups Algeb a Uni e salis 8(59-64), 1978 . [61 T ias,J . : Con ibución al es udio de los anillos e iculados y -ani-, llos . S ochas ica, ol III, no-2 (45-69), 1979 . [ 7 1 Vulikh,B .Z . : In oduc ion o he Theo y o Pa ially O de ed Spaoes . G oningen,1967 .