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Trees As Semilattices 1 Introduction

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Trees As Semilattices 1 Introduction Trees as Semilattices Leonid Libkin Department of Computer and Information Science University of Pennsylvania Philadelphia PA USA Email libkinsaulcisupennedu y Vladimir Gurvich RUTCOR Rutgers Center for Operations Research Rutgers University New Brunswick NJ USA Email gurvichrutcorrutgersedu Abstract We study semilattices whose diagrams are trees First we characterize them as semilattices whose convex subsemilattices form a convex geometry or equivalently the closure induced by convex subsemilattices is antiexchange Then we give lattice theoretic and two graph theoretic characterizations of atomistic semilattices with tree diagrams Intro duction Graph theoretic prop erties of lattice and semilattice diagrams are of great interest in lattice theory in combinatorics Even such fundamental prop erties of lattices as distributivity and mo dularity can b e expressed as prop erties of diagrams Various graph theoretic prop erties of diagrams give rise to very interesting classes of lattices For example planar lattices were characterized in via a numb er of forbidden congurations A simple forbidden conguration a p oset with the diagram like the letter N has a nice characterization for p osets which generalizes smo othly to lattices and semilattices In this pap er we lo ok at a very simple prop erty of a p oset diagram we study nite p osets whose diagrams are ro oted trees Such p osets are semilattices b ecause unique paths from any two no des to the ro ot have a minimal common p oint which is the least upp er b ound Chains b eing the only exception lattice diagrams are not trees but a similar investigation for lattices can b e carried out if only nonzero elements are considered However lattices whose nonzero elements have a tree diagram are equivalent to tree diagram semilattices The pap er is organized in three sections In the remainder of this section we give all necessary denitions In Section we characterize treediagram semilattices as semilattices having antiexchange closures induced by their convex subsemilattices Families of closed sets of antiexchange closures are known under the name of convex geometries and families of complements of closed sets are sometimes Supp orted in part by NSF Grant IRI and ATT Do ctoral Fellowship y On leave from the International Institute of Earthquake Prediction Theory and Mathematical Geophysics Moscow Russia referred to as antimatroids see It is wellknown that the closure op erator induced by subsemilattices of a semilattice is antiexchange If the family of subsemilattices is restricted to the convex ones then the antiexchange prop erty gives us treediagram semilattices In Section atomistic tree diagram semilattices are studied Three characterizations are obtained Firstly it is shown that such semilattices are exactly seriesparallel atomistic semilattices Secondly trees arising as diagrams of such semilattices are characterized as branchy trees ie trees whose ver tices except for leaves have at least two children Finally it is observed that treediagram semilattices can b e describ ed by complete chromatic graphs with four forbidden subgraphs In the sequel lattices and semilattices will b e denoted by the letters L and S resp ectively p ossibly with indices and and will stand for the least and the greatest elements In this pap er we consider only nite lattices and semilattices The semilattices are joinsemilattices that is the order is given by x y x y y Graphs will b e denoted by hV E i where V is a set of vertices and E a set of edges A tree with a ro ot s will b e denoted by hV E si A semilattice is called treediagram if its diagram is a ro oted tree with ro ot In the sequel we shall always assume that whenever the diagram of a semilattice is a tree it is ro oted and the ro ot is the maximal element This corresp onds to the denition of a computer science tree in In a poset tree is a p oset whose cover graph is a tree that is do es not contain a circuit Generally a p oset tree may not b e a computer science tree however in the case of nite semilattices these two denitions are equivalent Below all other denitions are given Treediagram lattice A lattice L such that the diagram of the joinsemilattice L fg is a tree Seriesparal lel poset A p oset containing no fourelement subp oset with diagram like the letter N Seriesparal lel lattice semilattice A lattice semilattice which is seriesparallel as a p oset Antiexchange closure A closure G on a set X satisfying A X x y X x y GA x GA y y GA x Convex geometry A family of closed sets of an antiexchange closure S ubS or S ubL The family of all subsemilattices or sublattices of S or L C S ubS The family of all convex or order preserving subsemilattices of S that is subsemilattices S such that x y z and x z S imply y S Ordinal sum of posets hP i and hP i with P P The p oset hP P i where 1 1 2 2 1 2 1 2 coincides with and on P and P and if p P p P then p p This p oset is denoted 1 2 1 2 1 1 2 2 1 2 by P P 1 2 Singleelement poset will b e denoted by and stands for a twoelement chain Branchy tree A ro oted tree hV E si such that v al s and for all v V s v al v where v al v j fw V v w E g j ie all vertices that are not leaves have at least two children Atomistic lattice A lattice every nonzero element of which is the join of atoms Atomistic semilattice A semilattice every element of which is the join of the minimal elements b elow it If L is atomistic then so is L fg considered as a joinsemilattice Treediagram lattices and semilattices and the antiexchange clo sures In this section we rst show that treediagram lattices are of form L S where S is a tree diagram semilattice Therefore all results ab out treediagram semilattices can b e reformulated for treediagram lattices in a straightforward manner Then we prove the main result of the section stating that a semilattice S is treediagram i C S ubS is a convex geometry We start with a simple lemma whose pro of is omitted Lemma Let S be a treediagram semilattice and S its subsemilattice with the least element x Then S is a chain In particular a lattice L is treediagram i L S for a treediagram semilattice S 2 Therefore it suces to prove all results for treediagram semilattices only Now we are ready to prove the main result of this section Theorem A semilattice S is treediagram i C S ubS is a convex geometry Proof Let S b e a treediagram semilattice To prove that C S ubS is a convex geometry we must show that for every S C S ubS if S S then there exists x S such that S x C S ubS see equivalent denitions of convex geometry in If S has unique minimal element it is a chain by lemma and its intervals form a convex geometry Supp ose S has two or more minimal elements Two cases arise Case S contains all minimal elements of S If S S then we are done If S S consider the top element y of S which do es not coincide with b ecause S is convex Then its lter y has at least two elements and since y is a subsemilattice by lemma y is covered by a unique element x Prove that S x C S ubS Clearly S x S ubS It is enough to prove that b S whenever a b x for a S Let c b e a minimal element of S such that b c Then c is a chain and y c Hence either b y or y b By the denitions of x and b y b Therefore b S b ecause S C S ubS Thus S x C S ubS Case There is a minimal element of S which do es not b elong to S Then there is an element x S covered by y S Prove that S x C S ubS Let z S Then z y z x Since x is a chain and y covers x we have that z x y and z x z y S Hence S S ubS Let x z v S Since x is a chain y is the unique cover of x and z y Since S is order preserving so is S x Therefore C S ubS is a convex geometry Conversely assume that S is a semilattice whose diagram S is not a tree Consider a circuit on this diagram Let x b e a minimal element of this circuit and y z its neighb ors Then b oth y and z cover x Let p y z Then p y z and the minimal order preserving subsemilattice containing fx pg or fx y z g is x p Then according to the list of the equivalent denitions of convex geometries C S ubS is not convex geometry b ecause in a convex geometry no set may have two dierent bases The theorem is completely proved 2 There is another relationship b etween tree diagrams and convex geometries if a lattice L is tree diagram then S ubL is a convex geometry Indeed a treediagram lattice is seriesparallel having an N would imply having a circuit and S ubL is a convex geometry i L is seriesparallel We have seen that in a treediagram semilattice two incomparable elements can not have a common lower b ound Therefore if x a a in a treediagram semilattice x a a for appropriate 1 n i j i j f ng Tree diagram lattices or semilattices are planar and hence have dimension one or two Either of these facts implies that treediagram lattices are distributive that is they satisfy x y y y x y y x y y x y y cf 0 1 2 0 1 0 2 1 2 The structure of mo dular and distributive treediagram lattices can b e easily describ ed Let M b e n an np oint pro jective line ie M f a a g where a a a a whenever i j n 1 n i j i j Prop osition A lattice L is modular and treediagram i L L L where L is either isomorphic 1 2 1 to M for some n or empty and L is a chain n 2 Proof The if part is obvious To prove the only if part let L b e mo dular treediagram lattice If jLj we are done Let jLj Let
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