On the Interplay b etween Interval Dimension and Dimension y z x S Felsner M Habib RH Mohring Abstract This pap er investigates a transformation P Q b etween partial orders P Q that transforms the interval dimension of P to the dimension of Q ie idimP dimQ Such a construction has b een shown b efore in the context of Ferrers dimension by Cogis Our construction can b e shown to b e equivalent to his but it has the advantage of b eing purely ordertheoretic providing a geometric interpretation of interval dimension similar to that of Ore for dimension and revealing several somewhat surprising connections to other ordertheoretic results For instance the transformation P Q can b e seen as almost an inverse of the wellknown split op eration it provides a theoretical background for the inuence of edge sub division on dimension eg the results of Spinrad and interval dimension and it turns out to b e invariant with resp ect to changes of P that do not alter its comparability graph thus providing also a simple new pro of for the comparability invariance of interval dimension This work was partially supp orted by the DFG and by the DAAD Pro ject PROCOPE y TUBerlin Berlin Germany z LRIMM Montp ellier France x TUBerlin Berlin Germany Intro duction An extension Q of a partial order P is an order on the same elements that contains all the ordered pairs of P ie x y in P implies x y in Q A family fQ Q g of extensions of P is said to realize P or to b e a realizer of P i 1 k P Q Q ie x y in P i x y in Q for each i i k If we 1 k i restrict the Q to b elong to a sp ecial class of orders and seek for a minimum size i realizer we come up with a concept of dimension with resp ect to the sp ecial class A linear extension of P is an extension which is a chain Dushnik and Miller dened the dimension of a partial order P denoted dimP as the smallest integer k for which there exist k linear extensions realizing P Let the real izer fL L g of an order P with jP j n b e made by the linear extensions 1 k L x x x With every x P we then asso ciate the vector i i1 i2 in 1 k k i x x R where x gives the p osition co ordinate of x in L This map i k ping of the p oints of P to p oints of R emb eds P into the comp onentwise ordering k k of R Ore dened dimP as the minimum k such that P emb eds into R in this way The pro jections of such an emb edding on each co ordinate yield a realizer of P The extensions of this realizer need not b e linear extensions of P but it is straightforward to transform them into linear extensions Therefore Ore dimension and DushnikMiller dimension are equivalent concepts A partial order P is an interval order if it can b e represented by assigning to each element x P an op en interval I a b of the real line such that x y x x x in P i b a Such a collection of intervals is called an interval representation of x y P Fishburn characterized interval orders as those orders that do not contain a p oint subset forming two disjoint chains ie that contain no Interval dimension denoted idimP is dened by using interval extensions instead of linear extensions Since linear orders are interval orders we obtain the trivial inequality idimP dimP It is well known that interval orders of large dimension exist see Hence the gap b etween idimP and dimP may b e arbitrarily large Let I fI I g b e an interval realizer of P and x an op en interval 1 k j j representation of each interval order I Let a b b e the interval corresp onding j x x to x P in the representation of I We now dene a box embedding of P j Q k j j k to R With x P we asso ciate the b ox a b R Each of these j x x k 1 and b b oxes is uniquely determined by its upper extreme corner u b x x x 1 k its lower extreme corner l a a Obviously x y in P i u l x x y x x comp onentwise The pro jections of a b ox emb edding onto each co ordinate yield an interval realizer so the concepts of b ox emb eddings and interval realizers are equivalent For interval dimension the b ox emb eddings thus play the role of the k ab ovementioned p oint emb eddings into R intro duced by Ore for dimension A b ox emb edding dep ends not only on the realizer I of P but also on the representations of the I Now we dene the partial order B I of extreme corners j asso ciated with a b ox emb edding or equivalently with an interval realizer I of P The vertices of B I are the at most n dierent lower resp upp er extreme corners of elements of P The order relation of B I is given by the comp onentwise order k in R k By denition we have an emb edding of B I in R so dim B I k idimP The starting p oint of our investigations was the following question concerning the interplay b etween dimension and interval dimension Is dim B I idimP In Section we dene a transformation P B P such that idimP dim B P We provide two interp etations of this transformation a combinatorial one and a geometrical one In the combinatorial interpretation the elements of B P are subsets of P In the geometrical interpretation B P is the p oset B I of extreme corners of I Here I is a b ox emb edding obtained from an arbitrary b ox emb edding I by a normalizing pro cedure From the pro ofs we obtain an armative answer to the question ab ove Section investigates several consequences to and relations with other order theoretic results First we study the transformation P B P on sp ecial partial orders of height In particular we show that the standard example S of a n n dimensional order is an almost xed p oint of the transformation P B P Therefore dimS idimS n n Second we investigate the relationship with the split op eration This has a 2 surprising consequence for the iterated transformation P B P B P k B P For every n there are partial orders P such that dimP k n+1 dim B P for all k n but dimP dim B P m where m is arbitrary Third we relate the interval dimension of sub divisions of P to the dimension of P thus providing a theoretical framework for the examples of Spinrad Finally we show the comparability invariance of the transformation P B P which as a consequence gives another pro of that the interval dimension is a comparability invariant Some remarks concerning the recognitioncomplexity of sp ecial classes of orders and graphs close the pap er The main result In the last section we dened the partial order B I of extreme corners asso ciated k with a b ox emb edding of P in R With the next lemmas we show that B I inherits some structure which is indep endent of the realizer I leading to the b ox emb edding Lemma Let B I be the partial order of extreme corners of a box representa tion of P a If the lower extreme corners of x and y are comparable in B I eg l x l then the predecessor sets of x and y in P are ordered by inclusion ie y Pred x Pred y P P b If the upper extreme corners of x and y are comparable in B I eg u u x y then the successor sets of x and y in P are ordered by reversed inclusion ie Succ x Succ y P P c If the lower extreme corner of x and the upper extreme corner of y are related by l u then Pred z Pred x for al l z Succ x or equivalently x y P P P Pred x Pred z P P z Succ (y ) P d If u l then Pred z Pred y x y P P z Succ (x) P j j for all j Therefore in each I x has a Pro of a From l l we obtain a j x y y x less predecessors than y ie Pred x Pred y The claim now follows from j j Pred x since the I realize P Pred x j j P j b The pro of of this part is symmetric to part a j j j j j c From l u we have a b If z Succ y then necessarily a b a x y x y z y x Hence Pred z Pred x for all j The claim follows j j d From u l we immediately obtain x y ie y Succ x therefore x y Pred y Pred z 2 z Succ (x) All statments except the conclusion part of b use only the sets Pred x and P Pred z This irregularity is resolved with the next lemma P z Succ (x) P Lemma Pred z Pred z if and only if Succ x Succ y z Succ (x) z Succ (y ) Pro of The if direction is trivial We now prove the only if direction Let Pred z From the assumed inclusion z Succ x and note that y z Succ (y ) we obtain y Pred z hence z Succ y 2 Denition With each vertex x of a partial order P we associate the lower set Lx Pred x and the upp er set U x Pred z The case x P P z Succ (x) P Max P is settled by the convention U x P Dene B P fLx U x x P g ordered by setinclusion Note that this construction is in fact equivalent with Cogis construction in the context of Ferrers dimension Cogis also uses Lx but replaces U x by the equivalent set fz P Succ x Succ z g He also proves Theorem but in a dierent way and without the geometrical interpretation that our approach is based on The preceding lemmas prove that l Lx and u U x together form x x an order preserving mapping from B I to B P hence idimP dim B I dim B P To get more structure into interval realizers we now intro duce a pro cedure that transforms a interval extension I f a b x P g of P into its P x x normalization I f a b x P g x x In the rst step of the P normalization we up date left endp oints a max fb z Pred xg if x is not minimal z x a minfa z MinP g if x is minimal z x In the second step we up date right endp oints b minfa z Succ xg if x is not maximal x z b max fb z Max P g if x is maximal z x Note that the interval order I need not b e isomorphic to I In general I is a sub order of I and a minimal interval extension