UNIVERSALITY OF S EPAROIDS .. 87 \ hspaceClearly∗{\ S openf i l l parenthesis}UNIVERSALITY M closing OF parenthesis S EPAROIDS = open\quad parenthesis87 E comma dagger closing parenthesis is a separoid period .. Furthermore comma the separoid is acyclic if and \noindentonly if theClearly oriented matroid $S is acyclic ( M period ) The = class ( of allE oriented , \ matroidsdagger is denoted) $ is a separoid . \quad 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, period†) 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 \ centerline { Figure 1 . \quad The acyclic s eparoids of order 3 . } 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 . \quad 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 $ \sigma s−e wit ˆ{ = }$ 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 $−hp s−mi { o−s−ts−ep } s−p ea−−ayt r−p ˆ{ vo−−starparenrighta r−i−e d−a }$ t $ e−sigma $ d $ f−i { t−r 1 .. e e-l mentsn s − t o−h p−m } rcolon−i −i{ o−t { s−p }} e−r e−r t−l { a−y t } p a r e n l e f t −i { v−star parenright −e } 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 ˆ{ l−l { e−e m−d a−e }}$ n $ s−t { colon−i }$ 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 .. \noindent w \quad i \quad e ( i \quad u \quad 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 B† B closinga n brace d C =† D openare brace m n comma− i im to the power of D \noindent $ i−s $ the supp o tof \quad aRa \quad donp a \quad riton;ie ,f $v−e $ er ys ubsetwi th \quad 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 ..\quad 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 $ \quad 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 iAB} = {, }. 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. \ hspaceforall x∗{\ negationslash-elementf i l l }Wes ay t a A ta cup\ Bquad existssepar Row 1 in $ Row o−dsi 2 $ A . di B sub sa :\ 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 { t−i dagger}$ 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 \noindent i s uni ue \quad $ i−n $ \quad it∈ su p\ or) $ semicolon−t $ th $ a−t ˆ{ i s , i f } ∀x 6∈ A ∪ B∃y ∪ B:(A y † (B \ y∪ x) A \dagger B $ a n \quad d $ C \daggerA D $ \quad are m $ n−i $ im \quad $ l−comma $ th n T he c − l s ofSteini z s p − e
\ [A \cup B = C {\cup } D = \Rightarrow \{ AB \} = \{ , ˆ{ D }\}{ . }\ ]
\noindent T \quad $ h−e c l a ˆ{ s−s }$ \quad o \quad Ra don \quad $ s−e $ p a r $ i−o { ds }$ i s \quad de $ n−o $ t d by $ R . $
\noindent s p a i $ S $ i a \quad S \quad $ i ˆ{ t }$ s p a $ i ˆ{ d }$ f i d $ ( S + 2 $ \quad l e $ m { e−n }$ t s , th $ n−e $
\ [ \ f o r a l l x \not\ in A \cup B \ exists y\ begin { array }{ c}\ in \\ A \end{ array }\cup B { :( } A ˆ{\setminus } y ˆ{ ) }\dagger{ ( } { B }\setminus y {\cup } x ) \ ]
\noindent T he \quad $ c−l $ s \quad o f S t e i n i z \quad s $ p−e $ 88 .. J period NE S-caron ET caron-R IL AND R period STRAUSZ \noindentFigure 2 period88 \quad .. SomeJ class . NE es of $ s\ eparoidscheck{ periodS} $ ET $ \check{R} $ IL AND R . STRAUSZ An oriented matroid is a Radon separoid whose minimal Radon partitions sat hyphen \ centerlineisfies the week{ Figure elimination 2 . axiom\quad : ifSome A sub class i dagger es B sub of i s comma eparoids for i = . 1} 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 { i }\dagger B { i } , $ 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 .. x titions for which there exists an $ x \ in B { 1 }\cap arrowdblright −comma { t−u n i v e r s a l −one−period−fo ur {\ f o r a l l }} negationslash-elementelimination C axiom cup D : ifperiodA †B A, for .. soi = served 1, 2, are b y minimal Radon par - titions for which there exists an x ∈ e−R−slash −notdef−J−T−brackrighti i −three −negationslash $ n $ element−three existential −notdef−t s V rgnasB brackleft∩arrowdblright 7 comma−comma o r iented m a-t roids .. a e S einitze − R s− paroidsslash − semicolonnotdef − J t− hT atis− brackright comma M− subsetthree R− capnegationslash double existential1 −period−e r−elementt−universal−negationslash−one−period−four∀ { e−b r a c e l e f t −T−notdef−four−negationslash existential −notdef stroke rightn arrowelement parenleft-element-five-one-hyphen−threeexistential−notdef−texistential e-arrowdblleft-one−period− er Z− element − negationslash e }$ i $ s−arrowdblleft $ ts $A$ amnim $ a−l $ R \quad $ d−a $e−braceleft on −T−notdef−four−negationslashexistential−notdefe gure 2 periodi s − arrowdblleft ts A a m nim a − l R d − a on rti−t i o n C † D s ch t at C ⊆ S A, D ⊆ S Bi d $ r t ˆ{ i−t }$ i o n $ C \dagger D $ s ch t at $ C \subseteq \bigcupa A,D naly commax 6∈ C ..∪ wD. saA y t-h so ata served .. e-s b paroidis y a g-r aph r-parenleft ecall E x-a sub m l-p e 3 comma i f fo rev ery m nim a-l \subseteq \bigcup B i { a }$ d \quad $ x \not\ in C \cup D . $ A \quad so served b y a-d on ps rt V to rgnas the [ power 7 , o r ientedof t-i io m n a A− daggert roids B a w e eh S einitz ve t at s paroids bar bar ; tbar h atis B bar, M = ⊂ 1 R∩ .. t ⇒ .. atiparenleft closing− parenthesiselement − five comma− one t ..− ehyphene .. − arrowdblleft − one m nim a-l Re d-aZ gure on 2 . \noindent sVrgnas [ 7 , o r ientedm $a−t $ r o i d s \quad a e S einitz s paroids ; t h atis nalyrt to , the w power sa y t of− h i-t ata io ns e ..− as e paroidis p irs o si a ngletonsg − r aph periodr − parenleft O s-b erve ecall t at E G x − subseta l − Rp period e 3 , i f fo rev ery m nim a − l $ , M \subset R \cap \Rightarrow p a r e n l e f t −elementm−f i v e −one−hyphen e−arrowdblleft −one $ 2 perioda ..− Td ..on heh p rt omomt−i io n o-rA phism† B w ehperiod-s ve t at S| .. ||nceB i-t|= s-i 1 .. e t ough ati t ) .., t k o e .. wm m nim nim a a-l− l R R d-a d − ona on p rt to the power of e $ Z $ i-hyphen rti−t io ns a e p irs o si ngletons . O s − b erve t at G ⊂ R. gure 2 . onsto ..2 r constructa. T heh .. fi omom ite s paroid o − r comma phism .. period w ec n− c ncentrates S nce io− ntts e− ..i s-t e udy ough .. t o th k e o .. m wm period nim a I− l rticular emdashR d − a w on h-e p rt ni− dhyphen finingonsto .. a no eration r constructa s-parenleft fi ite ee bs paroid lo w closing , w parenthesis ec n c ncentrate emdash o ntt is e en sough− t t-o .. d fine .. o-s \ centerline { naly , \quad w sa y $ t−h $ ata \quad $ e−s $ paroidis a $g−r $ aph $ r−p a r e n l e f t $ .. me udy o th e m . I rticular — w h − e n d fining a no eration s − parenleft ee b lo w ) — t is en e c a l l E $ x−a { m } l−p$ e3 , i f fo rev erymnim $a−l $ } rt to theough power t − ofo i-t io d nsfine .. a o d− ..s g nerate me rt ti− et ..io s ns paroid a d a .. g t neratee .. m nim t e l-a s syparoid .. mm a to t the e power m nim of l e-t− a ric fi ter c ntaining e .. g vensy s t mma ..e R−t a-dric on fi ter p rt c tontaining the power e g of ven t-i io s t to a the R power a − d of on e p ns rt periodt−iioe ns T .h-a T tish − ha comma tis h, w w ec ec n n d d fine .. a s paroid \noindent $ a−d $ on p $ r t ˆ{ t−i }$ i o n $ A \dagger B$ wehvetat $ \mid \mid S b yd finingfine a s paroid S b yd fining s − e t o ge neratorsof th e a m m t − e ric fi ter (, †, a , w h − e \mid B \mid = 1 $ \quad t \quad a t i $){ , }s $ t \quad e \quad m nim) $ a−l $ R $ d−a $ s-e t o gere neratorsofA † B C th† D ef a− subieA s m⊆ C .. m t-e ric fi ter open parenthesis comma dagger comma a sub preceq comma closing parenthesison w h-e re A dagger B preceq C dagger D f-i e A subset equal C $Equation: r t ˆ{ i d− ..t B}$ subset i o equal ns \ Dquad perioda e p irs o si ngletons .O $ s−b$ ervetat $G \subset Rt S . .. $ a d .. T b ew .. o s-e paroids period A m p-a pingfrB ⊆ omD. S t-o T s-i ca le d a h omom o-r phis m d th e e sub i m a-g e o .. m nim l-a R .. a-d on p rt to the power of t-i io ns .. a e m nim a-l R .. d-a on p rt to the power of i-t io\noindent ns periodt TS2 .. . a-ha\ dquad tisT commabT ew\quad o s −hehe paroids omom . A $ m o− p r− a $ pingfr phism om St $− periodo T s −−is ca $ le dS a\hquad omomnceo − r $ i−t s−i $ t−i \quad e oughphis m t th\quad e ei mk a − o g\ equad o mwm nim nim l − a $ R a− al− $d on R p rt $ d−ioa ns $ on a e mp nim $ ar t− l ˆ R{ i− dhyphen− a on }$ onsto \quadp rti−trio constructa ns . T a − h\ tisquad , $ fi $ ite s paroid , \quad w ec n c ncentrate o nt e \quad $ s−t $ udy \quad o th e \quad m . I r t i c u l a r −−− w $ h−e $ n d f i n i n g \quad a no eration $ s−parenleft $ ee b lo w ) −−− t i s en ough $ t−o $ \quad d f i n e \quad $ o−s $ \quad me $ r t ˆ{ i−t }$ i o ns \quad a d \quad g nerate t e \quad s paroid a \quad t e \quad m nim $ l−a $ sy \quad $ mm ˆ{ e−t }$ ric $ fi $ ter c ntaining e \quad g ven s t a \quad R $ a−d $ on p $ r t ˆ{ t−i } i o ˆ{ e }$ ns . T $ h−a $ t i s $h{ , }$ w ec n d f i n e \quad a s paroid $ S $ b yd fining $ s−e $ t o ge neratorsof th e $ a { s }$ m \quad m $ t−e$ ric $fi$ ter $( , \dagger , a {\preceq } , { ) }$ w $ h−e $ re $ A \dagger B \preceq C \dagger D f−i e{ A }\subseteq C $
\ begin { a l i g n ∗} \ tag ∗{$ d $} B \subseteq D. \end{ a l i g n ∗}
\noindent t $ S $ \quad a d \quad $ T $ b ew \quad o $ s−e $ paroids .Am $p−a $ p i n g f r om $ S t−o T s−i $ ca le dahomom $o−r $ phis m th e $ e { i }$ m $ a−g $ e o \quad m nim $ l−a $ R \quad $ a−d $ on p $ r t ˆ{ t−i }$ i o ns \quad a e m nim $ a−l $ R \quad $ d−a $ on p $ r t ˆ{ i−t }$ i o ns . T \quad $ a−h $ t i s , UNIVERSALITY OF S EPAROIDS .. 89 UNIVERSALITYa homomorphism OF is S a EPAROIDS mapping phi\quad : S right89 arrow T that satisfy for all A comma B subset equal S comma aA homomorphism dagger B minimal is equal-arrowdblright a mapping $ \varphi phi open parenthesis:S A\rightarrow closing parenthesisT$ dagger that phi satisfy open parenthesis for all B closing $A parenthesis,B \ minimalsubseteq S , $ open parenthesis as usual comma we put phi open parenthesis A closing parenthesis = open brace phi open parenthesis x closing \ [A \dagger B minimal \Longrightarrow \varphi (A) \dagger \varphi ( parenthesis semicolonUNIVERSALITY x in A closing OF S EPAROIDS brace closing 89 parenthesisa homomorphism period is a mapping ϕ : S → T that satisfy for all B ) minimal \ ] Two separoidsA, B ⊆ areS, .. is omorphic .. S thickapprox T if there is a bij ective homomorphism between them whose inverse function is also a homomorphism period If S subset T is a subset of a separoid comma the induced separoidA † Bminimal T brackleft =⇒ S brackrightϕ(A) † ϕ is(B the)minimal restriction to S period .. An embedding \noindentS right arrow( as T is usual an inj ective , we homomorphism put $ \varphi that is an( isomorphism A ) between= \{\ the domainvarphi ( x ) ; x \ in ( as usual , we put ϕ(A) = {ϕ(x); x ∈ A}). A and\} the induced) . separoid $ of its image period Two separoids are is omorphic S T if there is a bij ective homomorphism between them From now on comma we will denote by S right arrow≈ T the fact that there exists an homo hyphen whose inverse function is also a homomorphism . If S ⊂ T is a subset of a separoid , the induced Twomorphism separoids Also commaare \quad if S toi thes omorphic power of d38-d38\quad from$ Sthe separoid\ thickapprox S to the separoid T $ T ifcomma there and is by S a minus-arrowright bij ective homomorphism separoid T [S] is the restriction to S. An embedding S → T is an inj ective homomorphism that is Tbetween the other casethem period whose inverse function is also a homomorphism . If $ S \subset T$ is a subset of a separoidan isomorphism , the between induced the separoid domain and $Tthe induced [ separoid S ] of $ its is image the . restriction to $S . $ \quad An embedding $ S \rightarrowFrom now on , weT will$ denote is an by injS → ectiveT the fact homomorphism that there exists that an homo is - morphisman isomorphismAlso,ifSd38−d between38 the domain and thefrom induced the separoid separoidS to the of separoid its imageT, and . by S −→ T the other case .
From now on , we will denote by $ S \rightarrow T $ the fact that there exists an homo − $ morphism { Also , i f S ˆ{ d38−d38 }}$ from the separoid $ S $ to the separoid $T , $ and by $ S \longrightarrow T $ the other case . 9 0 .. J period NE S-caron ET caron-R IL AND R period STRAUSZ \noindentFigure 3 period9 0 \ ....quad ObserveJ . that NE P $ to\ thecheck power{S} of$ 2 is ET not a $ point\check separoid{R} $ open IL parenthesis AND R . nor STRAUSZ an oriented matroid closing parenthesis period \noindentFigure 3 periodFigure .. An 3 example . \ h f i wherel l Observe P in P but that P to $the P power ˆ{ 2 of} 2$ element-negationslash is not a point P separoid period ( nor an oriented matroid ) . I-m sub t m is easy to see that these constructions have the expected categorical proper hyphen \ centerline { Figure 3 . \quad An example where $ P \ in P $ but $ P ˆ{ 2 } element−negationslash ties : 1 S9 minus0 J . NE Sˇ ET Rˇ IL AND R . STRAUSZ P . $ } 1 periodFigure 3 period 3 . .. A .. comment .. on Observe .. Radon that quoterightP 2 is not s.. a pointlemma separoid period .. ( nor One an can oriented be tempted matroid to study ) . maps which preserve al lFigure Radon partitions 3 . An emdash example not where only theP ∈ minimal P but P ones2element period− ..negationslash We call suchP. \ hspace ∗{\ f i l l } $ I−m { t } m{ i s }$ easy to see that these constructions have the expected categorical proper − maps s trong morphismsI − periodmtmis .. easy In the tocategory see that these of separoids constructions endowed have with the strong expected mor categorical hyphen proper - phisms comma Radon quoteright s lemma can be formulated as follows open parenthesis cf period Lemma 6 closing parenthesis :\ begin { a l i g n ∗} t iTheorem e s : 1\\ open1 parenthesis S − Radon 1 92 1 closing parenthesisties : period S is a point to the power of a s eparoid to the power of in \end{ a l i g n ∗} of order bar S bar = d open parenthesis S closing parenthesis1S plus− 2 if and only if \noindentS minus-arrowright1 .1 3 . . 3A . K\ subquad comment 1 a-braceexA \quad on n-ccomment prime-d Radon S\quad ’ right s arrow lemmaon \ Kquad sub . 2RadonOne plus can sigma ’ be s commatempted\quad lemma to study . maps\quad One can be tempted to study maps whichw h-r sub preservewhich ere sigma preserve al A l isal Radona l sRadon a imp partitions to partitions the power —−−− of not l-i onlynot o m thei only d-S minimal sub the n s-F minimal ones sub . urt We ones h-p call sub such. ermore\quad maps subWes trong comma-R-braceleftmid c a l l such amaps in suc so-h trongmorphisms a cas e morphisms sub s-comma. In the . sigma category\quad S subIn of = separoids the plus-emptyset category endowed sigma withof iseparoids strong f-u-two mor an a-d - endowed on o-l sub with r-y l strong i i-f-y Case mor 1 i− closing brace Case 2 is inphisms , Radon ’ s lemma can be formulated as follows ( cf . Lemma 6 ) : \noindentg-t eneraTheorem l-yphisms p-comma 1 ,( Radon m Radon osi b-f-t 1921)’ s sub lemma.S ionis period a canpoint to be thea s formulated powereparoid of∈ uof asorder follows| S |= d ( (S cf) + 2 .if Lemma and only 6 ) if : a sub ndl e .. C = open brace c to the power of 0 comma period to the power of period period c closing brace sub period N o \noindent Theorem1(Radon $1 92 1 ) . S$ isa $pointˆ{ a }$ s $ eparoid ˆ{\ in }$ to the power of w comma l e K 2 toS the−→ powerK1 a of− =braceexn open brace− cprime b comma− d w S sub→ heK2 e+ aσ, dagger comma n d sigma = open brace c o fphi order open parenthesis $ \mid s-parenrightS \mid = =$d$( S ) + 2$ ifand w h − r σAis a s aimpl−i o mid − S s − F h − p ain suc o − h a onlyHowever i f the .. categoryere of separoids quoteright strongn morphismsurt seemsermorecomma to be− lessR−braceleftmid rich i } than thatcas of homomorphismse σS plus and− emptyset too restrictiveσif − u to− ourtwo purposesan a − d periodon o ..− Inl particularlii − f − y commaS − i \ [S \longrightarrows−comma = K { 1 } a−braceex n−c prime−r−dy S \rightarrowisin K { 2 } + \sigma , \ ] u g − teneral − yp − commamosib − f − tion.
\noindent w $ h−r 0{,...e r e }\w,lesigma= A{ i s }$ a s $a{ imp }ˆ{ l−i }$ o $m{ i } d−S { n } andl e C = {c c}. N o K2 { b, whe e a †, n d σ = {c s−F { urt } h−p { ermore { comma−R−braceleftmid }} a{ in }$ suc $ o−h $ a cas $ e { s−comma } \sigma S { = } plus−emptyset \sigmaϕ(s −{ parenrighti } f−u−two= $ an $ a−d $ on $\ l e f t . o−l { r−y } l { i } i−f−y S−i \ begin { a l i g n e d } & i \}\\ & i sHowever in \end the{ a l i g category n e d }\ right of separoids. $ ’ strong morphisms seems to be less rich than that of homo- morphisms and too restrictive to our purposes . In particular , \ begin { a l i g n ∗} g−t enera l−y p−comma m{ o s i } b−f−t ˆ{ u } { ion . } \end{ a l i g n ∗}
\noindent $ a { ndl }$ e \quad $ C = \{ c ˆ{ 0 , . ˆ{ . } . } c \} { . }$ N $ o ˆ{ w , l e } K 2 ˆ{ = }\{ b , w { he }$ e $ a \dagger , $ n d $ \sigma = \{ c $
\ [ \varphi ( s−parenright = \ ]
However the \quad category of separoids ’ strong morphisms seems to be less rich than that of homomorphisms and too restrictive to our purposes . \quad In particular , UNIVERSALITY OF S EPAROIDS .. 9 1 \ hspacesuch a∗{\ categoryf i l l } doesUNIVERSALITY not have a nice OF product S EPAROIDS period ..\ Forquad this9 comma 1 consider the separoids P sub 3 = open brace 0 comma 1 comma 2 closing brace where 0 dagger 1 2 comma .. and K sub 2 = open brace a comma b closing\noindent brace wheresuch a a dagger category b period does .. Let not us denote have a by nice product . \quad For this , consider the separoids $P P sub{ 3 times3 } K= sub 2\{ = open0 brace , 0 1 a comma , 0 2 b comma\} $ 1 a where comma 1 $ b 0 comma\dagger 2 a comma1 2 b closing 2 , brace $ \ thequad elementsand $ K { 2 } = \{ a , b \} $ where $ a \dagger b . $ \quad Let us denote by of the product and by pi and kappa the UNIVERSALITY OF S EPAROIDS 9 1 $ P { 3 }\times K { 2 } = \{ 0a,0b,1a,1b,2a two projectionssuch a category period If does A dagger not have B implies a nice that product pi open . parenthesis For this , consider A closing the parenthesis separoids P dagger= pi open{0, 1, parenthesis2} B closing , 2 b \} $ the elements of the product and by $ \ pi $ and $ 3\kappa $ the parenthesiswhere and kappa 0 † 12 open, and parenthesisK = A{ closinga, b} where parenthesisa † b. daggerLet us kappa denote open by P parenthesis× K = {0 Ba, closing0b, 1a, 1 parenthesisb, 2a, 2b} then the natural two projections . If $2 A \dagger B$ implies that $ 3\ pi 2 (A) \dagger \ pi ( candidatesthe to elements A and Bof are the A product = open and brace by π 0and a closingκ the brace two projections comma B .= If openA † B braceimplies 1 b that commaπ(A 2) † bπ closing(B) brace period .. B ) $ and $ \kappa (A) \dagger \kappa ( B ) $ then the natural However thisand wouldκ(A) imply† κ(B) then the natural candidates to A and B are A = {0a},B = {1b, 2b}. However this candidates to $A$ and $B$ are $A = \{ 0 a \} , B = \{ 1 b , that A daggerwould B imply cup open brace 0 b closing brace but comma pi open parenthesis 0 a closing parenthesis cap arrowdblright-zero four-period-one-universal-b2 b \} . $ \quad commaHowever universal-one this sub would b-negationslash-three-brackright-T-J-notdef-slash-R imply comma three-element- two sub b closing parenthesis period-existential Row 1 open brace Row 2 negationslash-four-notdef-T-braceleft . seven-existential existential-intersection\ begin { a l i g n ∗} arrowdblright-three-notdef-zero-notdef-T-d-notdef-union comma-hyphen-element-arrowdblleft to the power thatA † B ∪ {0b}but, that A \dagger B \cup \{ 0 b \} but , \\\ pi ( 0 a ) \cap arrowdblright −zero of element-one-approxequal sub comma two-four-period-negationslash-five-existential sub closing brace double stroke left arrow { 0 four−periodπ(0−a)one∩ arrowdblright−u n i v e r s a l −−bzerofour ,− universalperiod − one−one− universal{ b−negationslash− b, universal −−onethree −brackright −T−J−notdef−slash −R } , three − element − two )period − existentialnegationslash − element − equal period − universal − bracerightseven − existentialexistential − intersectionarrowdblright − three − notdef − zero − notdef − T − d − notdef − unioncomma − hyphen − element − arrowdblleftelement−one−approxequaltwo − four − period − negationslash − five − existential ⇐ braceleft − equal }= 6 φ. Row 1 0 Row 2 open brace . not = phi period b−negationslash−three−brackright−T −J−notdef−slash−R b negationslash − four − notdef − T − braceleft , } { ,h-e three refore− Pelement 3 sub times−two K 2{ subb s} ould) h ve period i-s ze 0− ..existential I no her w o-r ds negationslash comma w i-h l-e− ..element w r-o king−equal .. w th\ begin s rong{ array }{ c}\{\\ negationslash −four−notdef−T−b r a c e l e f t \end{ array } period−u n i v e r s a l −braceright seven−existential o-r ph i-Bh − s sube refore s m s-commaP 3×K2s pould sub h t eve o i p− t-o-rs ze duct 0 Isub no I her o t wt ese o − subr ds e s, paw i n-r− hl t oids− e r i w e re− uo o-i-s king valent w sub e t b t e i s m cexistential p-l a-o id periodth s− rongintersection o t o ..− h-ir ph s i − Bs arrowdblrightm s − commap −ethreeo p t−−notdefo − rduct−zeroo− tnotdef t ese s− paT− nd−−notdefrtoids r− iunion e e u comma−hyphen−element−arrowdblleft ˆ{ element−one−approxequal } { , } two−four−period−negationslashs −f i v e −existentialt {\}}\I Leftarrowe b r a c e l e f t −equal \ begin { array }{ c} 0 \\ llaps m-eo r− si e− issvalent t o be et o b c t h e urring i s m c sub p − mla e-t− o o id sub . o f ot ten h − subi s v llaps t r m m a-k− e e r period s e is t t o e b h e c o ncept c hur ˚ em toe − thetof power of o st rong sub\{\ lend d-m{ o-rarray ph i-e}\}\ Case 1not h Case= 2 y \phi . h \end{ a l i go n ten∗} t r m a − k e . t e h c ncept eo st rong d − mo − r ph i − e sm teroestinge. c l ter to the powerv of o esting to the power of e period cl l y e h .. e sub th to the power of a sub t i to the power of dagger j .. emdash c to the power of e sub ar comma K k to the power \noindent $ ha−e$†j reforee $Pi 3c {\timesg } K 2 { ns }$ ould h ve $ i−s $ ze 0 \quad I no her w of i sub s the e h to the etht poweri — of c car .. o, mp Kk etes th to e the powero mp eteof g ra phhe nce t Rowh 1 nt Row@on . R 2 e o e . c sub l l emm at o suba a n period R .. e c l l $ o−r $ ds , w $ i−h l−e $ \quad w $ r−o $ king \quade ow th s rong emm a a 2 period ed Row 1 d Row 2 he . ngs atem nt s i sub a reequival to the power of s e n t-colon sub d S o sub f or d to the $ o−r $2 . phe T $ i−olowiBngs s { atems } nt$ s mi reequival $ s−commas e n t − pcolon{ tSo}$or d ee r o pS |= $ t−do+− 1r a n duct d { I }$ o t t power of e r .. S bar =he d plus 1 a n .. d a d f $ ess period e { ae S} sub$ g s Row pa 1 $ d Rown−r 2 thyphen{ o i d os s-i}$ w m r a i . oe aru e u to the $ o power−i−s of h v m ale e subnt e{ t le n} shows$ tbteismc to the power of d sub$p t −l d l toa− theo $ power ids of. . a e oS pt h a\quad t .. n sub$ t hl se−oi taru s $ subh sm a ne dt to l n theshows powerdeph of a a t r ton these tpowers n of da ere uu u u Rowa 1y l Rowe 2o l te . e d sub n .. o te d y b g −os − iwma e t t a l n sigmal l a p period s $ m−e$ rseistobeoc $h{\ mathring { ur }} { m } e−t o { f }$ o $ ten { v }$ ybσ. 2. | S |= a d S p) + id. n t r2 mperiod $ abar−k$ S bar e.tehcncept = a d S p closing parenthesis $eˆ plus{ io d} period$ s t .. n $ rong { l } d−m o−r $ ph $ i−e \ l e f t . sm\ begin { a l i g n e d } & h \\Equation: 3 period S .. minus right arrow K open parenthesis sub 1 1 Equation: 4 period .. forall T negationslash-equal varnothing& y \ :end S minus{ a l i g right n e d arrow}\ right T period. $ − → K(1 1 3.S $In t the e r study ˆ{ o of} homomorphismse s t i n g ˆ{ e comma} . it $ is useful c l to have simple conditions which ∀T 6= : S− → T. 4. forbids them period .. The following is similar to the No∅ hyphen Homomorphism lemma open parenthesis cf period .. brackleft \noindent e h \quad $ e { th }ˆ{ a } { t } i ˆ{\dagger j }$ \quad −−− $ c ˆ{ e } { ar } 5 brackright closingIn the parenthesis study of homomorphisms period , it is useful to have simple conditions which forbids them . The ,Lemma K k 3 period ˆ{ i ..} Let{ s T} be$ a sth eparoid $ e in ˆ{ generalc }$ position\quad periodo mp .. If $ S eminus-arrowright t e ˆ{ g }$ raK sub phhe 1 .. nceand S t minus $h\ Tbegin { array }{ cc } n \\ e & o \followingend{ array is similar} t { to@ the oNo{ - Homomorphismn }} . $ Rlemma\quad ( cfe . c l [ 5 l ] ) . then d openLemma parenthesis 3 . SLet closingT be parenthesis a s eparoid greater in general equal position d open . parenthesis If S −→ T closingK and parenthesisS − T then periodd emmProof a aperiod 2 . For e the $\ l contrary e f t .T\ commabegin { supposearray }{ thatc} dd open\\ parenthesishe \end{ Sarray closing} olowi parenthesis1 \ right less.$ d open ngs parenthesis atem nt sT closing $ i { a } r e e q u i v a l(S ˆ){≥ sd (}T$). e n $ t−colon { d } S o { f }$ or $ d ˆ{ e }$ r \quad $ S \mid parenthesisProof open parenthesis . For the and contrary then d , opensuppose parenthesis that d (S S) closing< d (T parenthesis)( and then plus d (S 2) +less 2 ≤ ord equal (T ) + d 1)open. Since parenthesis T closing =parenthesis d + plus 1 closing $ a nparenthesis\quad d period S −→ K1, due to Lemma 2 , the order of S is at least d (S) + 2. Then there exists a minimal Since S minus-arrowrightRadon partition A K† subB such 1 comma that | ..A due∪ B to| Lemma ≤ d (S) 2 + comma 2. If theϕ : orderS → ofT Sis is a at function least d open, then parenthesis S closing parenthesis\noindent pluss 2. period a $ .. S Then{ g } t \ begin { array }{ c} d \\ − o s−i w m a \end{ array } l $ o $ aru ˆ{ h }$ m $ e {| ϕe(A}∪$B) t|l ≤ $nd{ (T )+1shows. Since}ˆ{ Td is}ˆ in{ generale } { positiont } p,T{ [hϕ(A}$∪B)] a∼ t K\1quadand ϕ(A$) isn separated{ t }$ se t $ s { a }$ there existsfrom aϕ minimal(B). Therefore Radonϕ partitionis not an A homomorphism dagger B such . that bar A cup B bar less or equal d open parenthesis S closing parenthesisn $ d ˆ{ plusa } 2 periodr ˆ{ ..e If }$ u u $a\ begin { array }{ c} l \\ l \end{ array }y e { n }$ \quad o te d $y{phib :}\ S rightsigma arrow T. is a$ function comma .. then .. bar phi open parenthesis A cup B closing parenthesis bar less or equal .. d open$ parenthesis2 . \ Tmid closingS parenthesis\mid plus= $1 period a d .. $ Since S T p{ is) in} general+ i { d } . $ \quad n position comma T brackleft phi open parenthesis A cup B closing parenthesis brackright thicksim K sub 1 and phi open parenthesis\ begin { a l A i g closing n ∗} parenthesis is separated from phi open parenthesis B closing parenthesis period Therefore phi is not \ tagan∗{ homomorphism$ 3 . S period $} − \rightarrow K( { 1 } 1 \\\ tag ∗{$ 4 . $}\ f o r a l l T \not= \ varnothing : S − \rightarrow T. \end{ a l i g n ∗}
In the study of homomorphisms , it is useful to have simple conditions which forbids them . \quad The following is similar to the No − Homomorphism lemma ( cf . \quad [ 5 ] ) .
\noindent Lemma 3 . \quad Let $ T $ be a s eparoid in general position . \quad I f $ S \longrightarrow K { 1 }$ \quad and $ S − T $ then d $ ( S ) \geq $ d $ ( T ) . $
\noindent Proof . For the contrary , suppose that d $ ( S ) < $ d $( T ) ($ andthend $ ( S ) + 2 \ leq $d$(T)+1 ) .$ Since $ S \longrightarrow K { 1 } , $ \quad due to Lemma 2 , the order of $ S $ is at least d $ ( S ) + 2 . $ \quad Then there exists a minimal Radon partition $ A \dagger B$ such that $ \mid A \cup B \mid \ leq $ d $ ( S ) + 2 . $ \quad I f $ \varphi :S \rightarrow T$ is a function , \quad then \quad $ \mid \varphi ( A \cup B) \mid \ leq $ \quad d $ ( T ) + 1 . $ \quad Since $T$ is in general position $, T [ \varphi (A \cup B)] \sim K { 1 }$ and $ \varphi ( A ) $ is separated from $ \varphi ( B ) .$ Therefore $ \varphi $ i s not an homomorphism . UNIVERSALITY OF S EPAROIDS .. 9 3 \ hspacea minimal∗{\ Radonf i l l }UNIVERSALITY partition iff ij in OF E semicolon S EPAROIDS .. and\ commaquad for9 3 each homomorphism of graphs phi : G right arrow H = open parenthesis V to the power of prime comma E to the power of prime closing parenthesis comma let\noindent Capital Psia open minimal parenthesis Radon phi partition closing parenthesis iff = $ phi ij period\ in .. ItE is easy ; to $ check\quad thatand Capital , for Psi is each an embedding homomorphism of graphs $emdash\varphi and in the:G sequel we\rightarrow often identify GH with =Capital ( Psi V open ˆ{\ parenthesisprime } G closing, E parenthesis ˆ{\prime period} ) , $ l e t $ \Psi ( \varphi ) = \varphi . $ \quad It is easy to check that $ \Psi $ is an embedding It is well known that the category of graphs endowed with homomorphismsUNIVERSALITY is OF S EPAROIDS 9 3 h universaa minimal m-l for counta Radon n-b partition le ca t-s iff ij sub egories∈ E; sub and semicolon , for each to thehomomorphism power of r c of u graphsi periodϕ e: periodG → H comma= every coun i-t \noindent −−− and in the sequel we often identify $ G $ with $ \Psi ( G ) . $ sub a s-b l-period(V 0,E0 sub), let e Ψ(categoryϕ) = ϕ. canIt be is embedded easy to check that Ψ is an embedding into the— category and in of the graphs sequel period we often This identify impliesG thatwith Ψ(G). It is well known that the category of graphs endowed with homomorphisms is AnalogouslyIt comma is well the known quasi that hyphen the category order open of graphsparenthesis endowed G comma with homomorphisms less or equal closing is h parenthesisuniversam comma− l where we put $h{ universa } m−l $ for counta $n−b $ l e ca $ t−s { e g o r i e s }ˆ{ r c } { ; } u{ i } G less or equalfor counta H whenevern−b Gle right ca t− arrowsr c H commaui. e . , every coun i−t s−bl − period category can be embedded . $ e . , every coun $i−t egories{ a; } s−b l−perioda { e }$ categorye can be embedded is universalinto period the category As an immediate of graphs . consequence This implies of that this fact comma we have that into the category of graphs . This implies that Theorem 5 period .... The homomorphismsAnalogously , the order quasi of S - is order universal (G, ≤) for, where countable we put ordersG ≤ periodH whenever G → H, Now commais universal let us denote. As an by immediate C to the power consequence of times of the this closure fact , in we S have of the that class C under products comma sums \ hspace ∗{\ f i l l } Analogously , the quasi − order $ ( G , \ leq ) ,$ whereweput $G and inducedTheorem substructures 5 . period .. ForThe example homomorphisms comma it order is well of hyphenS is universal known that for countable G = K to orders the power . of times open \ leq H$ whenever $G \rightarrow H , $ parenthesis seeNow brackleft , let 5 us brackright denote by closingC× the parenthesis closure in commaS of the class C under products , sums and induced w h-h subsubstructures ere e-K m-d . sub For enotes example t i-h sub , it e is c well u-l sub - known ass o s-fthat compleG = K to× the( see power [ 5 of] ) t-o , e grap to the power of e-h s parenleft-a \noindent is universal . As an immediate consequence oft− thiso facte−h , we have that see a l-r subw so h T-period− heree − heoremKm − 9denotes and Examplet i − he 7c closingu − lass parenthesiso s − fcomple period e grap s parenleft − a see a Analogousl − tor TheoremT − period 4 we heorem have 9the and following Example 7 ) . Analogous to Theorem 4 we have the following \noindent Theoremso 5 . \ h f i l l The homomorphisms order of $ S $ is universal for countable orders . g-r aphsg period− r aphs .. n . .. e n to the eP powerc i of P l c .. c e .. i g.. r l c .. s e ..u g r .. s .. u The category of separoids containsThe category other of rich separoids open parenthesis contains other and natural rich ( and closing natural parenthesis ) categories .. categories . We period .. We Now , let us denote by $ C ˆ{\times }$ the closure in $ S $ of the class $C$ under products , sums present herepresent another here anotherexample example : the category : the category H of hypergraphsH of hypergraphs open parenthesis( set systems set systems ) . Recall closing parenthesis period Recall and induced substructures . \quad For example , it is well − knownthat $G = Kˆ{\times } that a hypergraphthat a hypergraph H = openH parenthesis= (V,E) is aV set commaV endowed E closing with parenthesis a family of is subsets a set VE endowed⊆ 2V , with a family of subsets E subset ( $ see [ 5 ] ) , equal 2 to thecalled power edges of V , and comma that a homomorphism of hypergraphs H → H0, where H0 = (V 0,E0), is any map called edgesϕ : V comma→ V 0 which and that sends a homomorphism edges to edges .of hypergraphs H right arrow H to the power of prime comma where H to the \noindent w $ h−h { e r e } e−K m−d { enotes }$ t $ i−h { e }$ c $ u−l { ass }$ o $ s−f power of primeTheorem = 7 . There exists a functoral embedding Φ: H → S from the category of hypergraphs comple ˆ{ t−o }$ e $ grap ˆ{ e−h }$ s $ parenleft −a $ se e a $ l−r { so } T−period $ heorem 9 and Example 7 ) . open parenthesisinto that V of to s eparoids the power . of prime Therefore comma, ES to theis a power universal of prime partially closing or parenthesis - dered class comma ; that is any is , map phi : V right Analogous to Theorem 4 we have the following arrow V to thegiven power a partially of prime ordered which s sends e t (X edges, ≤), to edgesthere exists period a monotone Theorem 7 period .. There .. exists a functoral embedding Capital Phi : H right arrow S from the category \noindentof hypergraphs$ g− ..r into $ that aphs of s . eparoids\quad periodn \quad .. Therefore$ e ˆ comma{ P }$ S ..\ isquad a universalc \quad partiallyi \quad .. or hyphenl c \quad e \quad g r \quad s \quad u dered class semicolon that is comma given a partiallyembeddingι ordered s: eX t → open S : parenthesis X comma less or equal closing parenthesis \ hspace ∗{\ f i l l }The category of separoids contains other rich ( and natural ) \quad categories . \quad We comma .. there exists a monotone x ≤ y ⇐⇒ ι(x) − ι(y). embedding iota : X right arrow S : x less or equal y arrowdblleft-arrowdblright iota open parenthesis x closing parenthesis minus iota\noindent open parenthesisProofpresent . y closing hereLet Φ parenthesis another : H → S example periodbe the function : the which category assigns $to H each $ hypergraph of hypergraphsH = (V,E ()( set systems ) . Recall Proof periodsimple .. andLet Capital without Phi isolated : H right points arrow ) the S separoid be the functionS = (V which∪ E, † assigns), whose to minimal each hypergraph Radon partitions H = \noindentopen parenthesisarethat defined a V hypergraphcommaby : U E† closinge minimal $H parenthesis in = S if U (open⊆ V, V parenthesis e ∈ ,E and E simpleU = )e. $ and isThe without a function set isolated $V$ Φ is points injective endowed closing . parenthesis with a family the of subsets separoid$ E \ Ssubseteq =Let openϕ : parenthesisV → V20 ˆbe{ VaV homomorphism cup} E, comma $ dagger of hypergraphs closing parenthesis ( the image comma of edges whose are edges ) that sends the minimalhypergraph Radon partitionsH = (V,E are defined) to the by hypergraph : U daggerH e0 minimal= (V 0,E0 in). SThe if U mapping subset equalϕ induces V comma a function e in E on the \noindentand U =edges ecalled period which .. edges will The be function denoted , and Capital that again by a Phi homomorphismϕ : isE injective→ E0 and period therefore of ..hypergraphs Let we phi have : also V right a $ function H arrow\rightarrow V to the power ofH prime ˆ{\prime be a } homomorphism, $ where of $ H ˆ{\prime } = $ $hypergraphs ( V ˆ{\ openprime parenthesis} , the E image ˆ{\ ofprime edgesΦ(ϕ are):} V edges)∪ E closing→ ,$V 0 ∪ parenthesis isanymapE0. that $sends\varphi the hypergraph:V H = open\rightarrow parenthesis V comma ˆ{\prime E closing}$ parenthesis which sends edges to edges . To see that this function Φ(ϕ) is a separoid homomorphism Φ(H) → Φ(H0), observe that each to the hypergraph H to the power of prime = open parenthesis V to the power of prime comma E to the power of prime closing minimal Radon partition U † e is mapped to a minimal Radon parenthesis\noindent periodTheorem The mapping 7 . \quad phi inducesThere a\ functionquad exists on the edges a functoral embedding $ \Phi :H \rightarrow S $which from will the be denoted category again by phi : E right arrow E to the power of prime and therefore we have also a function of hypergraphs \quad into that of s eparoids . \quad Therefore $ , S$ \quad is a universal partially \quad or − Capital Phi open parenthesis phi closing parenthesispartition : V cupϕ(U E) † rightϕ(e). arrow V to the power of prime cup E to the power of prime perioddered class ; that is , given a partially ordered s e t $ ( X , \ leq ) , $ \quad there exists a monotone To see that this function Capital Phi open parenthesis phi closing parenthesis is a separoid homomorphism Capital Phi open parenthesis\ begin { a l H i g closing n ∗} parenthesis right arrow Capital Phi open parenthesis H to the power of prime closing parenthesis comma embeddingobserve that each\ iota minimal:X Radon partition\rightarrow U dagger e isS: mapped\\ tox a minimal\ leq Radony \ i f f \ iota ( x ) − \ iotapartition( phi y open ) parenthesis . U closing parenthesis dagger phi open parenthesis e closing parenthesis period \end{ a l i g n ∗}
\noindent Proof . \quad Let $ \Phi :H \rightarrow S $ be the function which assigns to each hypergraph $ H = $ $( V , E ) ($ simpleandwithout isolated points ) the separoid $S = ( V \cup E, \dagger ) , $ whose minimal Radon partitions are defined by $ : U \dagger e $ minimal in S if $U \subseteq V , e \ in E $ and $ U = e . $ \quad The function $ \Phi $ is injective . \quad Let $ \varphi : V \rightarrow V ˆ{\prime }$ be a homomorphism of hypergraphs ( the image of edges are edges ) that sends the hypergraph $ H = ( V , E ) $ to the hypergraph $ H ˆ{\prime } = ( V ˆ{\prime } , E ˆ{\prime } ) . $ The mapping $ \varphi $ induces a function on the edges which will be denoted again by $ \varphi :E \rightarrow E ˆ{\prime }$ and therefore we have also a function
\ [ \Phi ( \varphi ):V \cup E \rightarrow V ˆ{\prime }\cup E ˆ{\prime } . \ ]
To see that this function $ \Phi ( \varphi ) $ is a separoid homomorphism $ \Phi ( H) \rightarrow \Phi ( H ˆ{\prime } ) , $ observe that each minimal Radon partition $ U \dagger e $ is mapped to a minimal Radon
\ begin { a l i g n ∗} p a r t i t i o n \varphi (U) \dagger \varphi ( e ) . \end{ a l i g n ∗} UNIVERSALITY OF S EPAROIDS .. 9 5 \ hspaceTheorem∗{\ 8f period i l l }UNIVERSALITY .... If P right arrow OF Q S is EPAROIDS a gap comma\quad and Q9 is 5 not connected comma then there exists a gap S right arrow T with T connected period .. Furthermore comma Q thicksim T plus P and S thickapprox T times P : \noindentLine 1 Q thicksimTheorem T 8 plus . P\ h Line f i l l 2 d116-d116-d116-d116-d116-d116-d116I f $ P \rightarrow Q$ d74-d74-d74-d74-d74-d74-d74 is a gap , and $Q$ sub is forall not Line connected 3 T P , then there exists a gap Line 4 d74-d74-d74-d74-d74-d74-d74 to the power of exists d116-d116-d116-d116-d116-d116-d116 Line 5 S thickapprox T times P \noindent $ S \rightarrow T$ with $T$ connected . \quad Furthermore $ , Q \sim Proof period Let Q = T sub 1 plus times times times plus T sub k commaUNIVERSALITY where each OF T S sub EPAROIDS i is connected 9 5 period Clearly P T + P $ and $ S \ thickapprox T \times P : $ right arrowTheorem P plus T sub 8 . i right arrow If P → Q is a gap , and Q is not connected , then there exists a gap Q and thenS → PT thicksimwith T Pconnected plus T sub . i or Furthermore P plus T sub,Q i thicksim∼ T + P Qand periodS ≈ ..T Since× P Q: minus-arrowright P comma there exist to the\ [ \ powerbegin of{ a minus l i g n e s d a} TQ = T\ subsim i T + P \\ d116such− thatd116 T− minus-arrowrightd116−d116−d116 P− andd116 therefore−d116 P plus d74− Td74 arrowright-minus−d74−d74−d74 P− andd74 P−d74 plus T{\ thicksimf o r a Q l l period}\\ .. Finally comma letTP R \\ Q ∼ T + P d74−d74−d74−d74−d74−d74−d74 ˆ{\ exists } d116−d116−d116−d116−d116−d116−d116 \\ be such thatd P116 times− d116 T right− d116 arrow− d R116 right− d116 arrow− d T116 period− d116 .. Sinced74 P− rightd74 − arrowd74 − Pd plus74 − Rd right74 − d arrow74 − d T74 comma∀ then P thicksim S \ thickapprox T \times P \end{ a l i g n e d }\ ] P plus R TP or P plus R thicksim T period .. Therefore comma if to the power of minus T minus-arrowright R comma then R minus P and d74 − d74 − d74 − d74 − d74 − d74 − d74∃ d116 − d116 − d116 − d116 − d116 − d116 − d116 R minus T times P which \noindentconcludes theProof proof . period Let $Q = T { 1 } + \cdot \cdot \cdot +S ≈ T × {P k } , $ where each $ TAs{ usuali } comma$ is an connected order open parenthesis . Clearly X comma $ P less\rightarrow or equal closingP parenthesis + T is said{ i to}\ be denserightarrow if for all x$ comma y $Q$Proof and then . Let $PQ = T\1sim+ ··· +PTk, where + T each{ Tii is}$ connected or $ . P Clearly +P T→{P i+ }\Ti →simQ and Q . $ \quad Since in X comma if x less y then − $ Qthere\ existslongrightarrowthen aP z in∼ XP + suchTi thator PP x+ lessTi ,$∼ z lessQ. there ySince semicolonQ $−→ exist i periodP, there ˆ{ e period − exist }$ commas s a T a= a $T densei Tsuch order = that isT an−→{ orderPi and}$ without such thattherefore $ T P +\longrightarrowT −→ P and P + T ∼P$Q. Finally and therefore , let R be such $P that +P × T T → R\longrightarrow→ T. Since P $ and gaps period We now exhibit some dense classes of separoids period First observe− that comma due to $ PTheorem +P 8 T→ commaP\+simR when→ T,Q searchingthen .P $ for∼\P aquad gap+ R itorFinally isP enough+ R ∼ ,toT. let lookTherefore pairs $R$ of objects, if T S right−→ R, arrowthen TR − P and befor such whichR that T− isT connected× $P Pwhich period\ concludestimes theT proof\rightarrow . R \rightarrow T . $ \quad Since $ P \rightarrow PTheorem + R 9 periodAs\rightarrow usual P to , an the order powerT (X, of≤ times) , is $ said .. then is to dense be $dense period P if\sim for all x,P y ∈ +X, if Rx $ < y then there exists a orProof $ Pperiodz ∈ +X ....such LetR that S\ minussimx < T z minus-arrowright
\ [S \rightarrow S + ( P \times T) \rightarrow T \ ]
\noindent so , it is enough to prove that the opposite arrows does not exist .
Since $ T $ is connected and $ T \longrightarrow S , $ every homomorphism $ T − S + ( P \times T ) $ most be an homomorphism $ T \rightarrow P \times T $ which , followed by the $ project ˆ{\rightarrow }$ ion , would lead an homomorphism $ T \rightarrow P . $ \quad However , since $ P $ is in general position and d $ ( P ) > $ d $( T ) ,$ duetotheNo − Homomorphism Lemma 3 , such an homomorphism does not exist .
\ hspace ∗{\ f i l l }Now , every homomorphism $ S + ( P \times T) \rightarrow S $ induces to an homomorphism
\ begin { a l i g n ∗} \varphi :P \times \end{ a l i g n ∗} 9 6 .. J period NE S-caron ET caron-R IL AND R period STRAUSZ \noindentBut phi sub9 p6 prime\quad openJ . parenthesis NE $ \check alpha closing{S} $ parenthesis ET $ \check = phi{ openR} $ parenthesis IL AND p R to . the STRAUSZ power of prime times alpha closingBut parenthesis $ \varphi = phi{ p open}\ parenthesisprime A( times\alpha alpha closing) parenthesis = \varphi and phi sub( p p prime ˆ{\ openprime parenthesis}\times beta closing\alpha parenthesis) = \ =varphi phi open parenthesis(A \ Btimes times beta\ closingalpha parenthesis) $ and comma $ therefore\varphi we{ havep }\ also prime ( \beta ) = that\varphi (B \times \beta ) , $ therefore we have also 0 phi sub9 p 6 prime J . NEopenSˇ parenthesisET Rˇ IL AND alpha R . STRAUSZ closing parenthesisBut ϕ 0(α) =daggerϕ(p × phiα) sub = ϕ p(A prime× α) and openϕ parenthesis0(β) = ϕ(B beta× β), closing parenthesis \noindent that p p comma therefore we have also an obviousthat contradiction period Hence the homomorphism phi does not exists and we are \ [ done\varphi period { p }\prime ( \alpha ) \dagger \varphi { p }\prime ( \beta ) , \ ] Example 7 period .. Notice that P to the power ofϕp times0(α) † equal-negationslashϕp0(β), S period .. It is also easy to see that R to the power of times = R period .. It is an obvious contradiction . Hence the homomorphism ϕ does not exists and we are done . natural to ask if there is a quotedblleft closing parenthesis hyphen 5 period-eight 9 1 open parenthesis ni e quotedblright .. Example 7 . Notice that P× 6= S. It is also easy to see that R× = R. It is natural to prope\noindent rsubc asswran obvious to the power contradiction of a-h t-i sub e-c . h-s Hence g-a l-e the l s-n homomorphism e-e paroids period $ Or\varphi $ does not exists and we are ask if there is a “ ) − 5period − eight91( ni e ” prope rsubc asswra−ht−ie−ch−s g − al − e l s − ne − e doneeven .more concrete comma a nice proper subclass which generates all Radon separoids period paroids . Or even more concrete , a nice proper subclass which generates all Radon separoids . However comma .. it seems that such a nice subclass may not exist period .. For comma .. consider the However , it seems that such a nice subclass may not exist . For , consider the \noindentseparoid depictedExample in Figure7 . \quad 2 as anNotice example that of a separoid $ P ˆ{\ in thetimes class}\ R capne doubleS stroke . $ right\quad arrowIt universal-one-hyphen is also easy to see that separoid depicted in Figure 2 as an example of a separoid in the class R∩ ⇒ universal − one − element-five-period$ R ˆ{\times } period= R . $ \quad I t i s hyphenelement − five − period. Such a separoid is indeed a prime separoid ; i . e . , it cannot be naturalSuch a separoid to ask is indeedif there a prime is separoid a ‘‘ $semicolon ) − i period5 e period period− commaeight it cannot 9 1 be expressed ($ nie’’ as a product\quad prope rsubc expressed as a product of other two separoids . It is part of an infinite family of prime separoids , $ asswrof other ˆ two{ a− separoidsh t− periodi { e ..− Itc is part h−s of}} an infiniteg−a family l−e of$ prime l $ separoids s−n commae−e $ each paroids of . Or each of the form : S = {1, ..., n} and eventhe form more : S concrete = open brace , a 1 comma nice proper period period subclass period commawhich n generates closing brace all and Radon separoids . However1 dagger 2 , comma\quad 1 2it dagger seems 3 comma that 1 such 23 dagger a nice 4 comma subclass period period may period not exist comma . 1 period\quad periodFor period , \quad openconsider parenthesis the 1 † 2, 12 † 3, 123 † 4, ..., 1...(n − 1) † n. n minus 1 closing parenthesis dagger n period \noindent separoid depicted in Figure 2 as an example of a separoid in the class $ R \cap ObserveObserve that the that previous the previous family contains family contains an element an elementin each dimension in each dimension period .. . On On the other extreme , \Rightarrow u n i v e r s a l −one−hyphen element−f i v e −period . $ the otherwe extreme have that comma the family we have of that all separoids the family of of5 or all more separoids points of 5 or more points Such a separoid is indeed a prime separoid ; i . e . , it cannot be expressed as a product in the planein the in plane .... convex in positionconvex .... position open parenthesis( i . i e period . , e period where comma each element .... where is separated each element from is its separated from its of other two separoids . \quad It is part of an infinite family of prime separoids , each of complementcomplement closing parenthesis) are prime separoids are prime . separoids We do not period know .. whether We do there not know is a whether “ )hyphen there−five.89 ischarac- a .. quotedblleft closing theform $: S = \{ 1 , . . . , n \} $ and parenthesisterisation to the power of prime of hyphen-five separoids period . 89 characterisation of prime separoids period Theorem 9 implies that \ [ 1 \dagger 2 , 1 2 \dagger 3 , 1 23 \dagger 4 , . . . , 1 TheoremCorollary 9 implies that 10 . Let C be any class of non - empty s eparoids . If P ⊆ C then C× is . . . ( n − 1 ) \dagger n . \ ] Corollarydense 10 period . .. Let C be any class of non hyphen empty s eparoids period .. If P subset equal C then C to the power of times .. is In particular , the class of all non - empty separoids is dense . dense periodAt first glance it seems that P is responsible for the density of S; however we can find dense \noindent Observe that the previous family contains an element in each dimension . \quad On In particularsubclasses comma of the separoids class of which all non does hyphen not arise empty from separoids point separoids is dense period the other extreme , we have that the family of all separoids of 5 or more points At first glance( see the it seems following that Remark P is responsible ) . for the density of S semicolon however we can find dense subclasses of separoids which does not arise from point separoids bar − P |= 2 | bar − TST a n dR ad n part \noindentopen parenthesisin the see plane the following in \ h Remark f i l l convex closing parenthesis position period\ h f i l l ( i . e . , \ h f i l l where each element is separated from its bar-P bar = 2 bar bar-T S bar-bar sub T .. a n dR ad n par sub t \noindent complement ) are prime separoids . \quad We do not know whether there is a \quad ‘‘ existential−notdefexistential−period Equation:A† element-negationslash ⇐ T ≤ mi | A, B braceleft-T-notdef-four-negationslash|} nd A∩ ⇒=φ R − slash − notdef universal-period− J − T − existential-sevenbrackright − three exists− negationslashelement union-notdef-d- − three element − negationslashbraceleft − T − notdef − four − negationslashuniversalT-notdef-zero-notdef-three$ ) ˆ{ hyphen−f− i vperiodexistential e } double. stroke 89 $− leftseven arrow∃union .. A dagger− notdef double− d − strokeT − notdef left arrow− zero T less− ornotdef equal− mthree i bar⇐ A comma B bar sub closingcharacterisation brace n sub d A cap of doubleprime stroke separoids right arrow . = sub phi R-slash-notdef-J-T-brackright-three-negationslash element-three to the powere − ofa existential-notdef r ly d P − parenleft existential-period) = 2 T | −2 and P is in ge er al pos i − t i on . W e c n o − n w p − r \ centerlinee-a r ly d P-parenleft{Theorem closing 9 implies parenthesis thato e = m 2} T t barS minus < S 2 and+ PP is× in ge< er T al pos i-t i on period W e c n o-n w p-r o e m ..Remark t .. S less . S plus P times less TIf P is the separoid described in the previous proof , then P ∈ Z \ R. \noindentRemarkTherefore periodCorollary ....P Ifis P neither is 10 the . separoid\ aquad point separoiddescribedLet $C$ nor in the an oriented beprevious any matroid proof class comma nor of a non thengraph− P .empty in Z backslash s eparoids R period . \quad I f $ P \subseteqThereforeCorollary P isC neither $ 12then a . point $ separoidC ˆ{\times nor an oriented}$ \quad matroidi s nor a graph period denseCorollary . 12 period 1 . The only gap of S is K0 → K1, 1 period .. The only2 .gapThe of S only is K duality sub 0 right pair arrow of S Kis subK1 1→ comma= negationslash − arrowrightK0. \ centerline2 period .. The{ In only particular duality pair , of the S is K class sub 1 ofright all arrow non =− negationslash-arrowrightempty separoids isK sub dense 0 period . } At first glance it seems that $ P $ is responsible for the density of $ S ; $ however we can find dense subclasses of separoids which does not arise from point separoids
\noindent ( see the following Remark ) .
\noindent $ bar−P \mid = 2 \mid bar−TS \bracevert { T }$ \quad andRadn $ par { t }$
\ begin { a l i g n ∗} A \dagger \Leftarrow T \ leq m i \mid A,B \mid {\}} n { d } A \cap \Rightarrow = {\phi } R−slash −notdef−J−T−brackright −three −negationslash element−thre e ˆ{ existential −notdef existential −period }\ tag ∗{$ element−negationslash braceleft −T−notdef−four−negationslash universal −period existential −seven \ exists union−notdef−d−T−notdef−zero−notdef−thre e \Leftarrow $} \end{ a l i g n ∗}
\noindent $ e−a $ r l y d $ P−parenleft ) = 2 T \mid − 2$ and $P$ is in ge er al pos $ i−t$ ion.Wecn $o−n $ w $ p−r $
\ centerline {o e m \quad t \quad $ S < S + P \times < T $ }
\noindent Remark . \ h f i l l If $ P $ is the separoid described in the previous proof , then $ P \ in Z \setminus R . $
\noindent Therefore $ P $ is neither a point separoid nor an oriented matroid nor a graph .
\noindent Corollary 12 .
\ centerline {1 . \quad The only gap of $S$ is $K { 0 }\rightarrow K { 1 } , $ }
\ centerline {2 . \quad The only duality pair of $ S $ is $K { 1 }\rightarrow = negationslash −arrowright K { 0 } . $ } 98 .. J period NE S-caron ET caron-R IL AND R period STRAUSZ \noindentLine 1 hline98 line-line\quad hlineJ . line-line NE $ Line\check 2 hline{S} period$ ET period $ period\check hline{R} hline$ ILline-line AND R . STRAUSZ Table ignored! \ [ \Figurebegin 4{ perioda l i g n e .. d The}\ r interval u l e {3em K}{ sub0.4 1 right pt } arrowl i n K e − subl i n 2 e in P\ periodr u l e {3em}{0.4 pt } l i n e −l i n e \\ \Ir t-lscript u l e {3em a-f}{ sub0.4 o pt n-l} l o-d... w-j s i-lessequal\ subr u l e m-m{3em period-m}{0.4 pt e-T}\ d i-hr u a-e l e t{3em r-e l-e}{ sub0.4 s-ypt } u-t t-hl i n sub e − a-fl i n o-t e \ subend comma{ a l i g n e d }\ ] C-P sub98 oro l-f J . l-period NE Sˇ ET subRˇ aryIL AND p-one R . three-l STRAUSZ sub period chi to the power of j i-b sub s to the power of e v e to the power of th to the power of j a i .. less or equal j .. a n to the power of d l less or equal m More generally comma let us denote by floorleft-barline − timesline floorright-barline the− line number of minimal Radon partitions period Theorem 14 period .... Let Z be a Steinitz s eparoid... in general positionline comma− line .... and R a Radon \ vspaces eparoid∗{3 period ex }\ centerline .. Then {\ framebox[.5 \ l i n e w i d t h ] {\ t e x t b f { Table ignored!}}}\ vspace ∗{3 ex} Z minus R equal-arrowdblright bar-floorleft Z bar-floorright less or equal bar-floorleft R bar-floorright period Proof period .. Let phi : Z right arrow R be a homomorphismTable ignored! period .. If floorleft-bar Z floorright-bar greater floorleft-bar R floorright-bar .. then there are two \ centerline { Figure 4 . \quad The interval $ K { 1 }\rightarrow K { 2 }$ in $ P . $ minimal Radon partitions of Z commaFigure say A4 . sub iThe dagger interval B subK i1 comma→ K2 forin iP =. 1 comma 2 comma mapping into the same } minimal Radon partition of R alpha dagger beta comma say phi open parenthesis A sub i closing parenthesis = alpha and phi I t − lscripta − fon − l l o − dw − j s i − lessequalm−mperiod−me−T d i − ha − e t r − el − es−yu − tt − ha−fo−t, open parenthesis B sub i closing parenthesis = beta periodji ..−b Withth outj a d C − Porol − fl − period p − onethree − l.χ e v e i ≤ j a n ` ≤ m \ centerlineloose of generality{ I $ comma t−l s c we r i p may tary suppose a−f { thato there} n− existssl $ an l x in $ A o− subd 2 backslash w−j $ As sub $ 1 i− openl e s s parenthesis e q u a l { observem−m that period−m e−T }$ d $ i−Moreh a generally−e $ t, let $ us r− denotee lby−efloorleft{ s−y−}baru·−floorrightt t−h− bar{ athe−f number o−t of{ minimal, }}$ } x negationslash-elementRadon partitions B . sub 1 closing parenthesis period .. Since Z is in general position and it is Steinitz comma bar A sub 1 cup B subTheorem 1 bar = d 14 open . parenthesisLet Z closingbe a Steinitz parenthesis s eparoid plus in 2 general position , and R a Radon \noindentand there exists$ C− aP { oro } l−f l−period { ary } p−one three −l { . }\ chi ˆ{ j { i−b }} { s ˆ{ e }}$ v $ e ˆ{s eparoidth ˆ{ .j } Thena }$ i \quad $ \ leq j $ \quad a $ n ˆ{ d }\ e l l \ leq m $
\ hspace ∗{\ f i l l }More generally , let us denote by $ floorleft −bar \cdot f l o o r r i g h t −bar $ the number of minimal Radon partitions . Z − R =⇒ bar − floorleftZbar − floorright ≤ bar − floorleftRbar − floorright.
\noindentProofTheorem . 14Let . ϕ\:hZ f i→ l l RLetbe a $ homomorphism Z $ be a Steinitz . If floorleft s eparoid− barZfloorright in general− bar position > , \ h f i l l and $ R $ a Radon floorleft − barRfloorright − bar then there are two minimal Radon partitions of Z, say Ai † Bi, for i = 1, 2, mapping into the same minimal Radon partition of Rα†β, say ϕ(Ai) = α and ϕ(Bi) = β. \noindent s eparoid . \quad Then With out loose of generality , we may suppose that there exists an x ∈ A2 \A1( observe that x 6∈ B1). Since Z is in general position and it is Steinitz , | A1 ∪ B1 |= d (Z) + 2 \ [Z − andR there\ existsLongrightarrow a bar−floorleft Z bar−f l o o r r i g h t \ leq bar−f l o o r l e f t R bar−floorright . \ ]
\noindent Proof . \quad Let $ \varphi :Z \rightarrow R $ be a homomorphism . \quad I f $ f l o o r l e f t −bar Z floorright −bar > f l o o r l e f t −bar R floorright −bar $ \quad then there are two minimal Radon partitions of $ Z , $ say $A { i }\dagger B { i } , $ f o r $ i = 1 , 2 , $ mapping into the same minimal Radon partition of $ R \alpha \dagger \beta , $ say $ \varphi (A { i } ) = \alpha $ and $ \varphi (B { i } ) = \beta . $ \quad With out loose of generality , we may suppose that there exists an $ x \ in A { 2 }\setminus A { 1 } ( $ observe that $ x \not\ in B { 1 } ) . $ \quad Since $ Z $ is in general position and it is Steinitz $ , \mid A { 1 }\cup B { 1 }\mid = $ d $ ( Z ) + 2 $
\noindent and there exists a