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Interval Reductions and Extensions of Orders: Bijections To Interval Reductions and Extensions of Orders Bijections to Chains in Lattices Stefan Felsner Jens Gustedt and Michel Morvan Freie Universitat Berlin Fachbereich Mathematik und Informatik Takustr Berlin Germany Email felsnerinffub erlinde Technische Universitat Berlin Fachbereich Mathematik Strae des Juni MA Berlin Germany Email gustedtmathtub erlinde LIAFA Universite Paris Denis Diderot Case place Jussieu Paris Cedex France Email morvanliafajussieufr Abstract We discuss bijections that relate families of chains in lattices asso ciated to an order P and families of interval orders dened on the ground set of P Two bijections of this typ e havebeenknown The bijection b etween maximal chains in the antichain lattice AP and the linear extensions of P A bijection between maximal chains in the lattice of maximal antichains A P and minimal interval extensions of P M We discuss two approaches to asso ciate interval orders to chains in AP This leads to new bijections generalizing Bijections and As a consequence wechar acterize the chains corresp onding to weakorder extensions and minimal weakorder extensions of P Seeking for a way of representing interval reductions of P bychains wecameup with the separation lattice S P Chains in this lattice enco de an interesting sub class of interval reductions of P Let S P b e the lattice of maximal separations M in the separation lattice Restricted to maximal separations the ab ove bijection sp ecializes to a bijection whic h nicely complementsand A bijection between maximal chains in the lattice of maximal separations S P and minimal interval reductions of P M Mathematics Sub ject Classications A A Key Words Chains lattice of antichains bijection linear extension interval extension interval reduction weak order April Intro duction Overview Several lattices can be asso ciated to a partial order P and reect dierent asp ects of the structure of P The most prominent examples are the DedekindMacNeille LP completion of P the lattice of antichains AP and the lattice of maximal antichains A P M We consider families of chains in such lattices related to P namely families of chains that bijectively corresp ond to certain families of orders related to P Examples for such families of orders are interval extensions of P interval reductions of P and interval orders that are essential for P in particular essential weak orders See Section for the denition of essential In the sequel wegiveseveral suc h bijections We b egin with a classical result app earing in early work of Bonnet Monjardet Pouzet and Stanley It connects chains in the lattice of antichains AP of an order P with linear extensions Bijection There is a bijection between the linear extensions of P and maximal chains in AP Some years ago Habib et al gave a theorem which is similar in spirit It connects chains in the lattice of maximal antichains A P of an order M P with minimal interval extensions Bijection There is a bijection between the minimal interval extensions of P and maximal chains in A P M In Section we dene what it means for an interval representation or interval order to be essential for P or stronglyessential for P We re late these notions to extensions and reductions of P In Subsection we digress to prove a theorem ab out interval order extensions Restrict the ex tension lattice of an nelement oredered set to those extensions which are interval orders It is shown Theorem that the covers of this restriction are covers of the extension lattice ie corresp ond to the addition of a single comparability In Section wecharacterize classes of interval orders corresp onding bi jectively to chains in AP First Bijection relates these chains to essential weakorders This extends to a characterization of those chains corresp ond ing to weakorder extensions Prop osition and minimal weakorder exten sions Prop osition Bijection is a second bijection involving chains in AP It maps the chains to stronglyessential representations of interval extensions of sub orders of P This extends to a characterization of those chains corresp onding to stronglyessential interval extensions of P Prop o sition In Section we show that Bijection sp ecializes to Bijection when restricted to maximal chains in A P In particular we obtain a new and M more transparent pro of for Bijection If the order P is an interval order Bijection is the well known char acterization of interval orders by the sequence of the maximal antichains Another characterization of interval orders involving a linear order on the predecessor and successor sets of single elements motivates the denition of two new lattices asso ciated to an order P In Section we dene and study the lattice of separations S P and the lattice of maximal separations S P Interval orders are characterized by the prop erty that S P is a M M chain Wealsocharacterize the lattice S P forN free orders P M Bijection in Section relates chains in S P and essential represen tations of interval reductions of P In Prop osition wecharacterize a class of chains generating each essential interval reduction exactly once In Section we restrict Bijection to maximal chains of maximal sep arations Bijection is a bijection between maximal chains in S P and M maximal interval reductions of P Note that bijections and are both maximal chains in some lattice asso ciated to P to extremal interval relate extensions or reductions When restricted to interval orders b oth bijections reduce to a classical characterization of interval orders FinallyinSection we show that our bijections may help solving opti mization and counting problems The idea is to use dynamic programming suchthat the dep endency graph of the dynamic program is one of the lat tices AP A P S P orS P The time complexity of this approach M M dep ends on the size of the lattice In some cases this size is p olynomially b ounded and we obtain p olynomial algorithms eg maximal interval re ductions of an N free order can b e counted in quadratic time Basics An order P V consists of a nite set V of elements of P and the P relations of P whicharepairsx y of elements with x y The relation P is transitive and irreexive In some cases we will see a reexive order P relation P An order Q is an interval order if there is a pair l r of functions assigning to each element x V real numb ers l r on the real line so Q x x that l r for all x V ie l r is an op en interval and x y if x x Q x x Q and only if r l for all x y V The pair l r iscalledarepresentation x y Q of the interval order Q See Mohring for a go o d intro duction to interval orders Denition An interval reduction of an order P is an interval order Q on the same ground set such that x y implies x y for al l x y An Q P interval reduction Q of P is a maximal interval reduction if there is no interval reduction R between Q and P ie with Q R P An interval extension of P is an interval order Q on the same ground set such that x y implies x y for al l x y An interval extension Q of P Q P is a minimal interval extension if there is no interval extension R between P and Q ie with P R Q Interval Representations Throughout this pap er we work with integer representations of interval orders ie with representations in which all interval endp oints are non negativeintegers An integer representation l r such that there is a p osi tiveinteger K with l V r V f Kg is called a dense representa tion The number K is the magnitude of the representation The magnitude Q of an interval order Q is dened as the minimal magnitude of an interval representation of Q It is well known see that any interval order Q has a unique representation of magnitude Q this representation is called the canonical representation The canonical representation of Q is closely related to the lattice of maximal antichains A QofQ M Characterization A P is a chain i P is an interval order M The canonical representation l r ofQ can b e obtained from the chain A A of maximal antichains of P by dening l minfi x x k interval A g and r maxfi x A g Therefore the magnitude of an i x i order is just the number of maximal antichains of the order Another nice characterization of the canonical representation of Q is the following Given a representation l r of magnitude K such that l V f K g and r V f K Kg then K Q and l r is the canonical representation of Q Given a dense representation of an interval order Q let D fx V i Q r ig Each D for i K is an ideal downward closed set ie x D x i and y x implies y D ofQ hence we obtain a chain Q D D D D V K Q of ideals Symmetrically U fx V i l g is a lter upward closed i Q x set of Q and V U U U U Q K is a chain of lters Let M be the set of elements whose intervals contain i i i Obviously M is an antichain of Q and D M U is a partition i i i i of V Q Denition Let P V bean order and Q V bean interval P Q order on the same ground set A dense representation of Q is cal led an essential representation for P if D is an ideal of P and U is a lter of P i i for al l i K An interval order Q is cal led essential for P if Q admits an essential representation for P Note that if Q is an essential interval order for P then Q can b e an extension of P it can be a reduction of P or it can be neither extension nor reduc tion Note also that the conditions on the ideals and on the lters in the denition of essential representations are indep endentofeach other Tosee
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