2 Discretely Ordered Rings

7/28/2019 2 Discretely Ordered Rings

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2Discretely rdered ingsWhat 's c lonewe par t l y n t i l yco l lpL t teBr" t t now not what 's es is ted [JLr r rs

Addrc:;.t ttt the Utrc'o ()tt id

We start r ,rr tucly f mociels f ar i thrnet icn earnest y wri t ingclorvt. tomes in rp le x iomshatareo t ' rv io i rs lyrue l t N . The resu l t ingheory ,PA- , isc lescribecln Sect ion . ancl sour basetheory ' o whichwe shal l ateraclc lindLrc t ionx ioms.P,4- hassomevery n ice ea tu res h ichwe exp lo re n th ischapter . npart iculzrr e wi l l show in Sect ion .2 thart ny moclelol Pt l- nray beregarcledsAnencl-extensionf the starrdarclodel,and we ntroclLlcehenotionof a ), sentence, .rnclhow hatPA- is strongenottgh o proveal lt rue.I , sentences.The clualqLrest ion,hetherPA- or any reztst ' t l tableextension f PA- is strongenough o di.sprouell ulse I' sentencesurnsout tctbe of a rathercl i f ferent at l l re-as perhaps rophesicd y Burns!Th isw i l l be the sub iec t f thenex tchapter . )

2 . 1 T I _ I E X I O M S O F P AOur { i rs t ive ax ionts ta te hat the b inary r" rnct ions ancl zrreulssociat ive,commutzr t ive, nc lsat is fy he d is tr ibut ive aw.

Ax1 :Ax2Ax3:Ax4:Ax-5:

Yx, , z ( ( x y )+ z x * ( y+ z ) )Y x , y ( x * y : y + x )V x , ,z ( ( x ' ) ' z : x ' ( y ' z ) )V x , y ( x . y : y ' x )V * , y , z ( x ' ( y + z ) : X I I x ' z ) .

As wil l a lreacly e apparent,we wri teour - /o-formtt l i tsn the nAtttrz l l 'wAy, o or examl t le e wr i te . r f y ins teac lf +(x ,y ) . We a lsoe inp loy l lthe r.rsuaIbbreviat ions,mitt ingbrackets herepossible with the con-vention hat the nrLrl t ip l icut iorrperat ion s tnore [-r incl i r lgharr hat olaclc l i t ion). l ten we wil l omit the nrul t ip l icat iorrigrr l together ' .Ve alsoempltty he notat ionof the lastchapter or the closeclernls i I + I +. . . -F I ( i r s ) represent inghe nteger t .By Ax l i t c loes ot t r la t te rrow16


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Theuxiortt.s f PA- l 1this is bracketccl but foi ' sake of c lef in i terress e may zlssLtmel + l + . . . - F I i s m p l i c i t l yr a c k e t e d( . . . ( ( l + ) + l ) + " ' ) + l ) . )Q a n dI are ust the constz interms0, 1 respect ively.i rn i lar lyf n e N atrc l is av a r i a b l e ,h e n i x c l e n o te she te rm ( " ' ( ( t +x )+ . r ) + " ' +x ) ( r r x s ) fn )0 , a n c l l e n o te si f n : 0 ; a l s o . t r "l e n o te s' ' ' ( ( t ' x ) ' x ) x ) ( r r x s )i f n> 0 , anc l leno tes f rz :0 .The next wo axioms tate hat0 isan clent i tyor * anclazero or ' , andtha t is an c len t i tyor ' .

Ax6:Ax7:

V. r ( ( x 0 :x ) A (x 0 :0 ) )V x ( x ' 1 : x ) .

N is, of coLrrse,quippedwith a l inearorcler.This is expressecly thefol lowingerxioms:Ax8 : Vx ,y , z ( (xx


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l8 Discretely rdcrarl rittg.sBeforewe g ive he as t wo ax ioms e t us note ha t Ax l -Ax l3 i rn t t l vconverseso AxL l , Ax12 andshow ha t he z in Ax l3 is un ic lL re .F o r heconve rseo Ax l l supposex* 1y4 -z a r rc l l x l )Vx(x>0)

(The word discrete houldnot be confusedwith the word discreet. venthough heyboth have he sameLatin root l)The ax iomsAx l -Ax15 cons t i tu tehe theoryPA- . PA- is the 'basetheory ' hat we shal l assume olds n every 9.o-structr- l ree consic ler.(Lateron we wi l l adcl ncluct ionxioms o PA- to get he ul l PA or Peunoarithmetic.) he standzirdmodel N clearly satisfiesPtl- , as cloesanynonstandard odelof Th(N) s inceAxl-Axl5 are al l f i rs t-order. notherexamples givenby the 9o-structtveZ[X)* which s clefined s ol[ows:ZlXl is the r ing of polynomialsn one variatr le with coeif ic iehfsromZ. ZlXl is made nto an 9o-structure y clef in ing n orcler makingX' in f in i te lyla rge ' .Morespec i f i ca l ly , i tr ,o t . . , o , teV-wi th , , *0 . t l ienwec le f i n e r * a 1 X + . . . * a , ,X" )0 i f f a , , ) 0 . G iven any tw o p o l y n o rn i a l sp ,qZ[X] , we de f ine >q i f f p -11>0. Z lX l . i s the J f , r -s t ruc t l r ref a l lnon-negotiueolynornialsn ZIX), i .e., Z[X]- hzis lomain

{pez[x)l4x]F p=- t].I t is straightforwarcio check hatZ[X]* FPA- . \Xl* doesnot sat isfyTh(N) ,however , ince o r exarnp le FYx-Jy (Zy : .vV2y 1:x ) , whe easr fy : a , 1 * , X + . ' ' * a , , X " Z f x l * a n c l < 2 y 2 t r r * 2 u , X + " ' ) - 2 u , , X "X , t h e n 1 : a t : " ' o , , : 0 . l : O r , a n c l c - ro t h2 y ( X a n c l y + l ( X , s c - lZIX).FVy(2y XA2y+ I + X).There is a corresponclence-retweenmoclelsof Prl- ancl so-calleldiscretelyrdered ing.s,,vhich e nov/clescribe. r-rpposeF PA-. Dehne


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The u.riom,; l PA-R: NI2 l - where is heec lL r iva lcncc'c lu t ion

( , " b) - (c ,d ) i f l 'a + c l b + con i \ ,1) ( lxcrc isc. v 'cr i fy that th is is inc le c l an cc lLt iva lcncc.Wc mi tyc l e n o teh c c c l u i v a l c n c el a s s f ( u ,h ) b y [ n , b ] a n c l l c f i n e , ' , < o n R b y

[n. r]+ [c,11] [rr c:, -rd]lu bl . l , , dl : luc' btl bc: url ll ,n, r)< , ' ,dl i f f a + r l< b + c.

I t is easy o chec[ l u . 0 l

is an embeclcl ing f J4o-structures I---> sencling&1 onto the collection ofnon - r regn t i velements f R .

Conversely, f R is any d iscrete lyorc lerr ing anclM is the / . /n-structr - r recons i s t i ng f a l l non-negat i ve lemen ts f R, then MFPA- , and repeat ingthe aboveconstruct ionwe recoverour or ig ina l r ing R. This . iust i f ies al l ingPtl- the thcory ofthe non-negotiuepurt.sof cli.scrctely rdereclrings.

We concl r " rc leect ion2. l by showir rg hat for ^ny ue NIFPA- there areno e lements e twcer r r anc l *1 . The fo l low inq le f i r r i t i o r r i l l be usc f r " r l .pr r . r i . . r rnoN., et MF PA-. The rnocl i f tac l;ubtruc: t ior t perut ior t , - , isLGfined-onM bv

l 9

t t : h -L I L / -

0the Lrn ic lLrel v l F b i c : a

i t M E u { l ti n M s . t .

i f M F U ) b .lVe sha l l f tenwr i teu-b fo r a -b whet rwe knorv h t t t 7b .

r ! r-7-:----. '--A7-if 'g''ptf,'',til'r-_iiLcrMF '}A-.ThcnM F V . Y , ( y > x + l > x - Fl ) .


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20 DiscretelyorderecLingsP ro o f . f x , y e fu l F y x l e t z : y - , r s o y :x *z an dhence >0 ( fo r i fz ulby

p(X),-+ (u)e M.Show hat is well-defined nd s an embedding f -54-r-structlrres.onclude hatZIXI. is somorphico a substructuref M.Now let o be the sentence itp(i) of lnwhere tp s quantffier-free.how hat

N F o ) M E o ) Z [ X ] * r oand henceZlX l . FVr-Th(N) , the set o f a l l t rue /a-sentences f the form V, r rp(x )with V/(t) quanti f ier-free.2.4 Veri fy that any orclerecling R (i .e., an -Zn-structure at isfyingAxl-Ax13 andAx16) satisf ies

V x , y ( x < . y n y - t t 0 ) ;thus o define n orderon a ring R it issufficiento specify hepositive lernentslfR .2 ,5 Le t R be he i ngZ fX ,Y l l (X '1-ZYt ) , .e . ,R i s hepo lynomia li ngZ lX ,Y)modulo he equiva lence

p 6 , n - q 6 , Y ) i f f p - q : r ' ( X ' - z Y t ) f o rs o r n epo lynomia leZ[X,Y] .


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I nitiul .\egn'tcnt.rtn.cl ntl-exlcnsir.tnsShow hatR canLrc l iscrete lyrc lerecl .ler rce how hat

PA- lVx , (x t 2 f 7.2.6 Showhatal lorderclings re ntegralkrrnains,.e., at is fy

Vx ,Y(x ' l 0 -+ r : 0Vy :0 ) ,zrnd ave haracter is t iccro. .e . .

_1(ru:0)fo r eachne N. G ive an exampleo f an in tegra ldonra inof charac ter is t i cero thatczrnnotbe orderecl.2.7 Show thatZ[X]l(2X2- l) can lre orclered,but not cl iscretely rclerecl.2 . 8 S h o w h a t Z l X . Y , Z ) l ( X Z - Y = ) c a n b e c l i s c r e t e l yr d e r e dw i t h 0 < X < Y 0 th is ol lows rornAx1 and as imp lenc l t rc t ionn k. n-7---:--t'--+::r J - _ q V p n. + i I f n , I , / c eN a n dn : l ' l c t l - r e nA * l n : 1 . k .Proof We f ix ancl rrg lrey nclucat ionn k. For c:0, PA- l9: l_.QbyA x 6 . I f P A - l n : l . l c , k ' : k + l a n d n ' : n - l [ , t h e n P A - l l . k ' :L. ( [+ 1) l . L+ !_by Ax5 ancl Ax7; hence PA- | I . k ' : p+ !_by theindtrct ion ypothesis,encePA- | I . k ' : n ' by Lemma2.3. n

I f n , k e N w i t hn 1 k , t h e nP A - l n 1 k .Proo f .By Ax14PA-19 by Ax1l so PA F c>n as equired. n

Fora l l k e N P A - F V x (x < - ->x :O V x : 1V ' . . V . r k ) .Proof. By induct ion n k. For k:0 th is ol lows rom Ax15.Suppose

P A - V x ( ; < k - * : O V x : _ 1 V . ' ' . V x : & ) ;thenPA- FVx(x


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InitiuL segme ts und encl-exten,rion,s 23satistying l q e.s)ir e s ancl losecl nder hesuccessc'rrunction '(x) -r l ) suchA S

I , Ip e V[.Yl ' lp hesclegree k]fo r eachno l t -zero ceN ( thcsecuts are ck tsec l r rc le r but not unc ler . ) anc l

J , , : \q e Z[X ] ' lq < p * n for sonre i e N]for eachpo lynomia l eZ[x ) ' 'o f non-zerodegree theseare not evenc losec l nc ler+ t .

( c ) Any model MFPt I has a proper end-ex tens ion at is fy ingPA- , fo r g ivenlv lFPA- le t I i be the d iscre te ly rc lered ing assoc ia tec li th M, a ,nd e t R [X l haveorclergiven byl x l e1X . . . t u , ,X " )0 i f f u , , t0

f o r e a c h p o l y n o m i a lr * a ,X * . * a , ,X " R [X ]w i t h a , , ,1 , . , e , , e R a n d a , , * 0 .Then R[Xln, the structure onsist ingf the non-negat ivelements f R[X], is anrocle l f PA- and hecomposi t ionf thenatura l rnbeclc l inustvt.-.>---> 11*t

is an ernbedclingf tV1 nto a proper nitialsegment f R[X1..lD.rt','t-il 'l i l i f ; ls an to-term then Vx


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24 Discretelyctrclerecling.s '-.---;iormLrlas ecluivalento a I11 ormr"r lancl ice versa,eqrr ivalento Vl-10(i , r ) ancl -1Vy0(i , r) is equivalentMany usefLrl rrc lnterest ing ropert ies f nurnbcrssuch s :

sincelAt0G,; r ) sto ly - t ) ( i , t ) .turnout o be A

and M cN are 9o-structures,i , M < r N t i f f

'x - y exists ''x is rreducible 'a z < x ( z * y : x ) ;. r > A V u ' - x V < x l ( u ' u x ) ,

ancl o on. The importance f the c lass f A, , orrnulass that they areulssolutei th respecto end-extensions.l#+r.l-,.,


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Initiul segme t,\ uncL nd-exten.;icttt.sI f 0 ( i ) i s Vy < t ( i ) 01 ( t , y ) , i s o f complex i ty * 1 , anc laeM. then

A/t 0(a)(9 for a l lb


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26 DiscretelyorderedringsThiscloeseaveopen he possibi l i tyhatsome nicelyaxiomatizecl 'heoryextending A cor-r ldroveal l t ruesentences. e takeup thisc;r-rest ionnthe nextchapter.

Exercisesfor 2.22.9 Show that, for any 9o-formula 0(x, y) and for any -9,r-term (t),

1x< t(y)O(x, )and

-lVx < t(y)-l0(*, y)areequivalentn the predicate alculus.2.10 Find a closed o-term I and an9o-formula9(x) such hat

NFVx< t? (x )and

N FVx< t-10(x\(This athers i l ly exerc ises here as a warningagainst potent ia l ourceofrnistakes)2.17 If.Mc."N are 9a-strvcturesndNFPA- show hatMFPA-.2.L2 Verify the fol lowingvariationon t he Tarski-Vaught est for elementarysubstructures:If Mc.N are substructureshen Tvll ttN iff for al l aeM, tor al l 9a-formulas0(x,!) e A0and or al l 9a-terms (t), NFSx


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Initialsegmets uncLnd-exte sons,(a) Exhib i ta rnodel f Q in which is not nterpretec l(b) Given t l ra t tv lFQ, ver i fy that the map n- l ! r 'J4,q-structures--+M with imagean initialsegment f lvl.Th ( N )

2.15 The fo l lowing xior lancl x ior lschemeslef inehe

2las a l inearorder .

i s an ernbedding ofF I e n c e e d u c eQ F E , --Sl*-theoryS:

SO:51e:SZp.;,S3e:S41.1 :551:561.1 :57p:

where k, I range over(a) Show that for al l

0 + 1 = l- l 0 : k + Ik + I : ! + 7 - -k : 1k + 0 =L + U - + 1 ) : ( & + 0 + 1k . 0 : 0& . Q _ + r ) : ( k . ! _ ) + kYx(x

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2 Discretely Ordered Rings