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 a a b e atoms of L n Supp ose a a a If

1 n 1 n

there is bka then by lemma bka and b a b a b ecause a is an atom Since b a and

1 1 1 1

a can not b e incomparable in view of lemma b a a This means that b a b a Therefore

1 1

f a a b a bg is a sublattice of L isomorphic to N This contradiction shows that L a L

1 5 2

where L is a chain by lemma

2

If n then L is a chain and L is empty If n we have to prove that a M To do this

1 n

we only have to show that a covers a for all i n Supp ose there is such i that a do es not cover

i

a ie a b a for some b Since b a there is a b Let x a b Then x a and by

i i j j i

lemma x and y a a are comparable If y were less than b we would have a b Hence y b

i j j

and therefore x y It shows that f a a b xg is a sublattice isomorphic to N This contradiction

i j 5

proves that a covers a ie a M and L M L where L is a chain Prop osition is proved

i n n 2 2

2

Corollary A lattice L is distributive and treediagram i L L L where L is either empty or

1 2 1

isomorphic to or and L is a chain 2

2

Atomistic treediagram semilattices and chromatic graphs

In this section we characterize atomistic treediagram semilattices as atomistic seriesparallel semilat

tices and show that their diagrams are branchy trees Then we extend the representation technique for

p ositional structures in game theory cf to describ e such semilattices via complete chromatic

subgraphs with four forbidden subgraphs

Let K hA E i b e a nite complete graph without lo ops and multiple edges ie E fa a

1 2

a a A a a g Let c E N b e a coloring mapping Usually N f ng ie edges are

1 2 1 2

colored with n colors ca a N is the color of the edge a a E Such a triple hN A ci

1 2 1 2

is called a chromatic graph Each subset A A generates a chromatic subgraph of In what

follows four chromatic subgraphs 2 j depicted on the gure b elow will pay the crucial role

@ @ @

@

@ @

@

@ @

@

@ @

@

@ @

@ @ @

2 j

Given a semilattice S let A b e the set of its atoms Dene a chromatic graph asso ciated with S

S S

as follows hS A c i where c x y x y S

S S S S

Theorem Given a semilattice S the fol lowing are equivalent

S is atomistic and seriesparal lel

S is atomistic and treediagram

The diagram of S is a branchy tree with root

In addition if S is atomistic then it is treediagram i the chromatic graph does not contain

S

subgraphs isomorphic to 2 j

Proof Let S b e atomistic and seriesparallel Prove that S is treediagram rst Assume

it is not and consider a circuit with a minimal element x and its neighb ors y z Both y and z cover

x Since S is atomistic there is an atom a y such that a z and hence a x Therefore

a y y x x z a y b ecause y is not an atom and akx akz y kz Thus S is not seriesparallel

This contradiction shows that S is treediagram Show that the diagram of S considered as a ro oted

tree with ro ot is branchy Supp ose there is an element x with v al x that is x covers a

unique element y b ecause x is covered by a unique element by lemma Consider an atom a x

such that a y Clearly x a for x is not an atom b ecause atoms are terminal vertices of the

considered ro oted tree and for every atom b v al b Hence there exists z a x covered by x

and since x covers only y z y Thus y a which contradicts our assumption Hence v al x

for all x If v al then let x b e the only element covered by and let a b e an atom There

exists an element y covered by in a and since x is the only element covered by x y a

Therefore x is greater than the join of all atoms and can not b e represented as the join of atoms

This contradiction shows v al and nishes the pro of of

Let hold Then S is treediagram and we must prove that S is atomistic Let x b e

a joinirreducible element which is not an atom Then x covers a unique element If x then

v al and if x then by lemma x has a unique cover and v al x ie the diagram of

S is not branchy This contradiction shows that S is atomistic

That implies follows from the fact that any treediagram semilattice is seriesparallel

To prove the last statement we need a few auxiliary denitions Let G hT N i where T is a

ro oted tree hV E si whose set of leaves is denoted by A N is a nite set and is a map from V A

to N These constructions are called p ositional structures in game theory Asso ciate a chromatic

graph G hN A ci with G where the coloring function is dened as follows If a a is an

i j

edge in let p b e the common no de of paths sa and sa which is farthest from the ro ot Then

ij i j

ca a p For example if T is a twocolored balanced binary tree of depth whose ro ot is

i j ij

colored by one color and intermediate no des by the other then applied to it would yield a chromatic

graph isomorphic to 2 We call G nonrep eated if b b whenever b b is an edge in T It was

proved by the second author in that the mapping is a corresp ondence b etween nonrep eated

structures G whose underlying trees are branchy and chromatic graphs without subgraphs isomorphic

to and

Moreover if is injective then G do es not contain a subgraph isomorphic to 2 or j Conversely

if is chromatic graph not containing subgraphs isomorphic to 2 and j by the result cited ab ove

there exists a unique nonrep eated structure G whose underlying tree is branchy such that G

Prove that of that structure G is injective Supp ose it is not that is b b for b b Since

G is nonrep eated b and b are not adjacent Then there exists a no de a inside the path bb such

that a b Two cases arise dep ending on whether there is a path from the ro ot containing

b oth b b It is easy to show that in the rst case when such path exists G contains a chromatic

subgraph isomorphic to j and in the second case when there is no such path G contains a chromatic

subgraph isomorphic to either j or 2 Therefore we have proved that the mapping establishes a

corresp ondence b etween nonrep eated structures G with injective functions and whose underlying

trees are branchy and chromatic graphs without chromatic subgraphs isomorphic to 2 j

Now given a treediagram semilattice S consider G S hT S A idi where T is the

S S S S

diagram of S The semilattice S is treediagram i T is branchy Therefore since S the

S S

onetoone corresp ondence established ab ove nishes the pro of of the theorem 2

Corollary For any treediagram semilattice S the chromatic graph does not contain subgraphs

S

isomorphic to 2 j Moreover the mapping S is a correspondence between atomistic

S

treediagram semilattices and chromatic graphs without subgraphs isomorphic to 2 j 2

Acknowledgements The authors would like to thank an anonymous referee for several helpful

suggestions

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