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On the Fractional Dimension of Partially Ordered Sets

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On the Fractional Dimension of Partially Ordered Sets On the Fractional Dimension of Partially Ordered Sets STEFAN FELSNER Bel l Communications Research South Street Morristown NJ USA and Technische Uni versitat Berlin Fachbereich Mathematik Strasse des Juni Berlin Germany Email address felsnermathtub erlinde WILLIAM T TROTTER Bel l Communications Research South Street L Morristown NJ USA and Depart ment of Mathematics Arizona State University Tempe AZ USA Email address wttb ellcorecom Date Septemb er Abstract We use a variety of combinatorial techniques to prove several theorems concerning fractional dimension of partially ordered sets In particular we settle a conjecture of Brightwell and Scheinerman by showing that the fractional dimension of a p oset is never more than the maximum degree plus one Furthermore when the maximum degree k is at least two we show that equality holds if and only if one of the comp onents of the p oset is isomorphic to S the standard example of a k dimensional k +1 p oset When w the fractional dimension of a p oset P of width w is less than w unless P contains S If P is a p oset containing an antichain A and at most n other p oints where n we show that the w fractional dimension of P is less than n unless P contains S If P contains an antichain A such that all n antichains disjoint from A have size at most w then the fractional dimension of P is at most w and this b ound is b est p ossible Mathematics Sub ject Classication A C Key words Partially p oset dimension fractional dimension degree Intro duction In this pap er we consider a partial ly ordered set P as a pair X P and refer to X as the ground set and P as the partial order We nd it convenient to use the short form poset for a partially ordered set and we use the term subposet to refer to a p oset induced by a subset of the ground set Let P X P b e a p oset and let F fM M g b e a multiset of linear extensions 1 t of P Brightwell and Scheinerman call F a k fold realizer of P if for each incomparable pair x y there are at least k linear extensions in F which reverse the pair x y ie jfi i t x y in M gj k The fractional dimension of P denoted by fdimP is i then dened in as the least real numb er q for which there exists a k fold realizer F fM M g of P so that k t q it is easily veried that the least upp er b ound 1 t of such real numb ers q is indeed attained Using this terminology the dimension of P denoted by dimP is just the least t for which there exists a fold realizer of P It follows immediately that fdimP dimP for every p oset P In this pap er we will nd it convenient to use the following probabilistic interpretation of this concept Let P X P b e a p oset and let F fM M g b e a multiset of 1 t linear extensions of P We consider the linear extensions of F as outcomes in a uniform sample space For an incomparable pair x y the probability that x is over y in F is FRACTIONAL DIMENSION given by fi i t x y in M g Prob x y i F t The fractional dimension of P is then the least rational numb er q so that there exists a multiset F fM M g of linear extensions of P with Prob x y q for 1 t F every incomparable pair x y For each n the height two p oset containing n minimal elements fx x x g 1 2 n and n maximal elements fy y y g with x y if and only if i j for all i j 1 2 n i j n will b e denoted by S The p oset S is known as the standard example of a n n ndimensional p oset As noted in fdimS dimS n n n This pap er is organized as follows In Section we intro duce some terminology nec essary for our theorems and review facts ab out lexicographic sums In Section we formulate the results relating the fractional dimension of a p oset to its degree In Sec tion we present a key technical lemma involving the algorithmic transformation of linear orders into linear extensions and present the pro of for the main theorem of Section In Section we present the forbidden subp oset characterization of the degree inequality In Sections and we derive two other forbidden subp oset characterizations of inequalities for fractional dimension In b oth cases equality is only p ossible when P contains a full dimensional standard example as a subp oset In Section we present an inequality relating the fractional dimension of a p oset P X P to the width of the subp oset induced by X A where A is an antichain The b ounds obtained in this case are slightly stronger than the corresp onding b ounds for dimension We also prove that our b ounds are b est p ossible In Section we characterize Hiraguchis inequality for fractional dimension Finally in Section we prop ose some new problems for fractional dimension Notation and Terminology Let P X P b e a p oset We denote the set of incomparable pairs of P by incP Now let F fM M M g b e a nonempty multiset of linear orders on X We call F a 1 2 t multirealizer of P if P F ie each M is a linear extension of P and prob x y i F for every x y incP When F is a multirealizer of P we dene the value of F denoted value F as the P least rational numb er q so that prob x y q for every x y incP The F fractional dimension of P is just the least rational q so that P has a multirealizer F of value q Recall that an incomparable pair x y in a p oset P X P is critical if any p oint less than x is less than y and any p oint greater than y is greater than x We denote the set of all critical pairs of P by critP The following elementary prop osition rst noted by Rabinovitch and Rival is stated formally to emphasize the fundamentally imp ortant role played by critical pairs in dimension theory PROPOSITION Let P X P be a poset and let F be a multiset of linear exten sions of P Then F is a multirealizer of P if and only if prob x y for every F x y critP We say that a linear order L reverses the critical pair x y if x y in L Similarly we say L reverses the set S crit P if L reverses each pair in S The following remark is noted in S FELSNER AND W T TROTTER PROPOSITION Let P X P be poset and let q be a rational number Then fdimP q if and only if there exists a multiset F of linear extensions of P so that Prob x y q whenever x y critP F Let P X P b e a p oset and let D fY Y Q x X g b e a family of p osets x x x indexed by the p oint set X of P The lexicographic sum of D over P is the p oset S whose p oint set consists of all pairs of the form x y where x X and y Y The partial x x in S if and only if x y order on S is dened by setting x y 2 x 1 x 2 1 x x in P or 1 2 in Q y x x and y x x 1 2 x 1 2 1 Both dimension and fractional dimension are well b ehaved with resp ect to lexicographic sums The statement for dimension is due to Hiraguchi and the statement for frac tional dimension is given in PROPOSITION Let S be the lexicographic sum of D fY x X g over P x X P Then dimS maxfdimP max dimY g xX x fdimS maxffdimP max fdimY g xX x A p oset S is decomposable with resp ect to lexicographic sums if it is isomorphic to a lexicographic sum of a family D fY x X g over a p oset P X P where jX j x and jY j for at least one x X Otherwise we say S is indecomposable x Given a p oint x in a p oset P X P let D x denote the set of all p oints which are P less than x in P and U x the set of p oints which are greater than x in P Whenever P the meaning is clear from the context we will drop the subscripts and just write D x and U x Two p oints x and y are said to b e interchangeable in a p oset P X P if D x D y and U x U y Thus x and y are interchangeable if and only if b oth x y and y x are critical pairs Note that a p oset with an interchangeable pair is decomp osable with resp ect to lexicographic sums unless it is a element antichain Fractional Dimension and Degree For a p oint x in a p oset P X P dene the degree of x denoted deg x as the numb er of p oints comparable but not equal to x in P Then let P denote the maximum degree of P ie P maxfdeg x x X g The following theorem is proved by Brightwell and Scheinerman in THEOREM If P X P is a poset then fdimP P In the next section we prove the following improved b ound which was conjectured in THEOREM If P X P is a poset and is not an antichain then fdimP P As noted in fdimS n S for all n so Theorem is b est p ossible n n In Section we show that when the the maximum degree of a connected p oset P is at least the inequality in Theorem is strict unless P is the standard example of dimension P This result has much the same avor as Bro oks theorem which asserts that the chromatic numb er of a connected graph is at most one more than the maximum degree with equality holding if and only if the graph is a complete graph or an o dd cycle It also provides another instance in which the role played by standard examples FRACTIONAL DIMENSION in dimension theory is analogous to the role played by complete graphs in the theory of graph coloring This parallel is explored in greater detail in the monograph We will in fact prove a result that is slightly stronger than Theorem Given a p oint x in a xed p oset P X P let D x D x fxg Then let deg x jD xj and D dene P maxfdeg x x X g The set U x and the quantities deg x P D U D U are dened dually With this notation we can now state the result we actually prove THEOREM If P X P is a poset then fdimP P D Note that Theorem has Theorem as an immediate corollary Since fractional dimension is dual ie a p oset and its dual have the
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