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Hspace ∗1< Fill Luniversality of S EPAROIDS <Quad 87 UNIVERSALITY OF S EPAROIDS .. 87 n hspaceClearly∗fn S openf i l l parenthesisgUNIVERSALITY M closing OF parenthesis S EPAROIDS = opennquad parenthesis87 E comma dagger closing parenthesis is a separoid period .. Furthermore comma the separoid is acyclic if and nnoindentonly if theClearly oriented matroid $S is acyclic ( M period ) The = class ( of allE oriented , n matroidsdagger is denoted) $ is a separoid . nquad Furthermore , the separoid is acyclic if and onlyby M if period the oriented matroid is acyclic . The class of all oriented matroids is denoted by $ M . $ 5 period All acyclic separoids of order 3 come from one of the eight familiesUNIVERSALITY of convex OF S EPAROIDS 87 bodies depictedClearly inS(M Figure) = 1(E; periody) is Those a separoid labelled . a Furthermore comma b comma , the e separoid and h are is point acyclic separoids if and only period if the 5 . All acyclic separoids of order 3 come from one of the eight families of convex Figure 1oriented period .. matroid The acyclic is acyclic s eparoids . The of class order of all3 period oriented matroids is denoted by M: bodies depicted in Figure 1 . Those labelled a , b , e and h are point separoids . Example 1 suggest5 . All the acyclic following separoids definitions of order period 3 come The from dimension one of the d open eight parenthesis families of convex S closing bodies parenthesis depicted of a separoid is the minimumin Figure number 1 . Those d such labelled that everya , b subset, e and withh dare plus point 2 elements separoids is the . support n centerline f Figure 1 . nquad The acyclic s eparoids of order 3 . g of a Radon partition period .. EquivalentlyFigure 1 . commaThe the acyclic dimension s eparoids of a ofseparoid order 3 is . the maximum d such thatExample there exists 1 suggest a subset the sigma following wit to definitions the power .of The = hdimension d plus 1 elementsd (S) such of a that separoid every is subset the min- of Example 1 suggest the following definitions . The dimension d $ ( S ) $ of a separoid i-A t s-iimum sub s-e number s-p e-ad r-psuch to that the power every ofsubset o-a r-i with d-ad t+ e-sigma 2 elements d f-i is sub the t-r support o-h p-m of r-i a Radon sub o-t partition sub s-p e-r . e-r t-l sub a-y t is the minimum number $ d $ such that every subset with $ d + 2 $ elements is the support parenleft-i subEquivalently v-star parenright-e , the dimension c-i o-s ofm-c a l-p-aseparoid to the is the power maximum of l-l subd e-esuch m-d that a-e there n s-t exists sub colon-i a subset σwit= of a Radon partition . nquad Equivalently , the dimension of a separoid is the maximum w .. i ..h e opend + 1 parenthesis elements such i .. u that .. a everye p r e subset e n of i − A t s − i s − pe − ar − po−ar−id−a t e − sigma d $ d $ such that there exists a subset $ nsigma s−e wit ^f = g$ h $ d + 1 $ elements such that every subset of i-s the suppf − i o tof .. aRar − ..i donp ae − ..re r it− onrt − semicolonl parenleft ie comma− i f v-e er ys ubsetwic − io th− ..sm d− parenleft-Scl − p − al closing−le−em−da parenthesis−e plus $ i−A $ tt−ro $−hp s−mi f o−s−ts−ep g s−p ea−−ayt r−p ^f vo−−starparenrighta r−i−e d−a g$ t $ e−sigma $ d $ f−i f t−r 1 .. e e-l mentsn s − t o−h p−m g rcolon−i −if o−t f s−p gg e−r e−r t−l f a−y t g p a r e n l e f t −i f v−star parenright −e g i-n du cw s-e .. i asi e mp ( i oid u perioda e p r e e n c−i o−s m−c l−p−a ^f l−l f e−e m−d a−e gg$ n $ s−t f colon−i g$ Wes ay ti − a tas the .. separ supp o-i o tof di sa aRa .. Radon donp separoid a r it f-i on ev ; erym ie , f vi im− e a er R ys ad ubsetwi o-n p art th sub d t-i parenleft ions − S) + 1 is uni uee .. e − i-nl ments .. it su i − pn or du semicolon-t c s − easi th a-t mp to oid the . power of is comma if A dagger B a n .. d C dagger D .. are m n-i im .. nnoindent w nquad i nquad e ( i nquad u nquad a e p r e e n l-comma th n Wes ay t a ta separ o − i di sa Radon separoid f − i ev erym i im a R ad o − n p artt−i ions A cup Bis = uni C sub ue cup i − Dn = itdouble su p stroke or semicolon right arrow− t th open a − bracetis;if A A By B closinga n brace d C =y D openare brace m n comma− i im to the power of D nnoindent $ i−s $ the supp o tof nquad aRa nquad donp a nquad riton;ie ,f $v−e $ er ys ubsetwi th nquad d closing bracel − periodcomma th n $ pT a .. r e h-e n l e cla f t − toS the )power + of s-s 1 .. $ o ..nquad Ra done .. s-e $ ep− al r $i-o sub ments ds is .. de n-o t d by R period $ i−n $ du c $ s−e $ nquad a s i mp oid . D s p a i S i a .. S .. i to the powerA of t[ s pB a i= to theC[ powerD = of) d f f iABg = f; g: d open parenthesis S plus 2 .. le m sub e-n ts comma th n-e T h − e clas−s o Ra don s − e p a r i − o is de n − o t d by R: n hspaceforall x∗fn negationslash-elementf i l l gWes ay t a A ta cupn Bquad existssepar Row 1 in $ Row o−dsi 2 $ A . di B sub sa :n openquad parenthesisRadon separoid A to the power $ f− ofi backslash $ ev erym y to i im aRad s p a i S i a S it s p a id f i d (S + 2 le m ts , th n − e the$ o− powern $ of p closing $ ar parenthesis t f t−i daggerg$ openi o n s parenthesis sub B backslashe−n y sub cup x closing parenthesis T he .. c-l s .. ofSteini z .. s p-e nnoindent i s uni ue nquad $ i−n $ nquad it2 su pn or) $ semicolon−t $ th $ a−t ^f i s , i f g 8x 62 A [ B9y [ B:(A y y (B n y[ x) A ndagger B $ a n nquad d $ C ndaggerA D $ nquad are m $ n−i $ im nquad $ l−comma $ th n T he c − l s ofSteini z s p − e n [A ncup B = C f ncup g D = nRightarrow nf AB ng = nf , ^f D g ngf . g n ] nnoindent T nquad $ h−e c l a ^f s−s g$ nquad o nquad Ra don nquad $ s−e $ p a r $ i−o f ds g$ i s nquad de $ n−o $ t d by $ R . $ nnoindent s p a i $ S $ i a nquad S nquad $ i ^f t g$ s p a $ i ^f d g$ f i d $ ( S + 2 $ nquad l e $ m f e−n g$ t s , th $ n−e $ n [ n f o r a l l x nnotn in A ncup B n exists yn begin f array gf cg n in nn A nendf array gncup B f :( g A ^f nsetminus g y ^f ) g ndaggerf ( g f B g nsetminus y f ncup g x ) n ] nnoindent T he nquad $ c−l $ s nquad o f S t e i n i z nquad s $ p−e $ 88 .. J period NE S-caron ET caron-R IL AND R period STRAUSZ nnoindentFigure 2 period88 nquad .. SomeJ class . NE es of $ sn eparoidscheckf periodSg $ ET $ ncheckfRg $ IL AND R . STRAUSZ An oriented matroid is a Radon separoid whose minimal Radon partitions sat hyphen n centerlineisfies the weekf Figure elimination 2 . axiomnquad : ifSome A sub class i dagger es B sub of i s comma eparoids for i = . 1g comma 2 comma are minimal Radon par hyphen titions for which there exists an x in B sub 1 cap arrowdblright-comma sub t-universal-one-period-four sub forall e-R-slash- An oriented matroid is a Radon separoid whose minimal Radon partitions sat − notdef-J-T-brackright-three-negationslash88 J . NE Sˇ ET Rˇ IL AND R .n STRAUSZ element-three existential-notdef-t existential-period-e r-element-negationslash sub e- isfies the week elimination axiom : if $ A f i g ndagger B f i g , $ f o r $ i = 1 braceleft-T-notdef-four-negationslash existential-notdefFigure 2 . Some e i s-arrowdblleft class es of s eparoidsts A a m . nim a-l R .. d-a on , 2 , $ are minimal Radon par − rt to the powerAn oriented of i-t i o matroid n C daggeris a D Radon s ch t separoid at C subset whose equal minimal union Radon of A comma partitions D subset sat - isfies equal the union week of B i sub a d .
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