AN AXIOMATIC APPROACH TO DISTANCES

BETWEEN CERTAIN DISCRETE STRUCTURES

T. Margush

A Dissertation

Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

August 1980 il

ABSTRACT

The need for a quantitative tool to measure distances between various kinds of discrete structures is apparent from the literature.

Distance concepts can also be used to judge the "goodness" of a candidate for the consensus of a collection of structures. Several axiomatic characterizations of distance functions on collections of binary relations and sets of subsets are presented. Applications to tree structures, preference relations, hierarchical classifications, cladistic characters, and partitions and other structures resulting from non-hierarchical classification techniques are discussed. The median of a collection, with respect to an appropriate distance function, is proposed as a candidate for consensus and methods for determining the median in the various contexts are presented. Ill

ACKNOWLEDGMENT

I would like to express my appreciation to Dr. F. R. McMorris for his continued interest in the progress of my research and his many suggestions and assistance in the writing of this dissertation.

I would also like to thank Linda Shellenbarger for her expertise in typing. IV

tS&e

TABLE OF CONTENTS

Page

CHAPTER 0 INTRODUCTION...... 1

CHAPTER 1 DISTANCES BETWEEN BINARY RELATIONS .... 4

CHAPTER 2 DISTANCES BETWEEN PARTITIONS AND OTHER SETS OF SUBSETS...... 37

CHAPTER 3 A CONSENSUS FOR DISCRETE STRUCTURES. ... 65

BIBLIOGRAPHY 78 CHAPTER 0

INTRODUCTION

In 1951, K. Arrow [2] listed several "reasonable" axioms which

should be satisfied by social welfare functions. The axioms proved

to be inconsistent, that is, there was no social welfare function

that satisfied all of the axioms. Much work has since been done,

revising Arrow's axioms in order to obtain characterizations of

various social welfare functions. In 1962, Kemeny and Snell [18]

gave another approach to the problem of determining "good" social y welfare functions. They gave an axiomatic characterization of a

distance function on the collection of choices and then examined

properties of medians with respect to the resulting distance function.

K. Bogart [5,6] generalized the work of Kemeny and Snell and showed

that in many instances, majority rule resulted in a median for

certain collections.

In numerical taxonomy, and other areas of science, there is

considerable interest in the comparison of hierarchical classifications

of collections of objects. Several types of "distance functions" have been proposed and their relative advantages discussed [10,13,15,20,

27,28,30]. An axiomatic development of both distance concepts and

consensus techniques on the various structures involved is an

-1- 2

important step in the attempt to quantify some of the subjective

analysis which takes place in these areas. Similar questions also

arise in the area of non-hierarchical classification techniques,

notably in cases concerning partitions of sets [8,12,22].

Chapter 1 begins with a summary of a portion of K. Bogart's

results found in [5], which is an extension of that of Kemeny and

Snell [18]. These results are then generalized to arbitrary collec­

tions of binary relations on a finite set J. The seven axioms

listed by Bogart are eventually replaced by a pair of axioms which

are equivalent to Bogart's on the collections of binary relations

considered by him.

Chapter 2 deals with similar concepts for sets of subsets of j/.

It first parallels the development in Chapter 1, then, following

along the lines of a paper by E. L. Perry [22], examines distance

concepts for collections of partitions that generalize to a new

distance function on sets of subsets. Several representations of

sets of subsets are introduced and their relative advantages discussed

in light of particular applications.

The last chapter examines the question of consensus for the various discrete structures of Chapters 1 and 2. First, Bogart's

results for collections of preference relations [5,6] are extended to arbitrary collections of binary relations and sets of subsets.

Majority rule is shown to be a median in each case (with respect to the distance functions characterized in Chapters 1 and 2) and questions of existence and uniqueness are examined. 3

For sets of subsets and the extension of Perry's distance function, a different consensus is defined. With the proper representation, it is shown that this may also be viewed as a type of majority rule. Questions of existence and uniqueness are then examined in this context. CHAPTER 1

DISTANCES BETWEEN BINARY RELATIONS

The use of binary relations to model preferences on a collection of objects has been widely accepted in the behavioral sciences [5,

26,33,34]. If ji is a finite collection of objects and a and b are elements of j/, then we may represent the preference of a to b by the ordered pair, (b,a). An individual's "preferences" among the elements of if can then be represented by a binary relation, R, on jf, where (b,a) £ R means the individual prefers a to b.

Not all binary relations on if can be reasonably interpreted as someone's preferences. For example, {(a,a)} is a binary relation on jf = {a,b}. The definition of "preference" tacitly assumes the comparison of at least two distinct objects, and therefore, this relation could not be the representation of an individual's preferences in if.

If a binary relation, R, on if, is to represent preferences, the pair (a, a) should not be an element of R for any a in ft.

That is, R should be an irreftexive relation on if. There are other commonly accepted properties of the relation "is preferred to" that can be easily interpreted in the language of binary relations. If a,b € / and a is preferred to b, then b cannot also be preferred to a. Thus, if (b,a) € R, then (a,b) £ R or R is an

-4- 5

antisymmetric relation. Preference is usually thought to be transitive: if a is preferred to b, and b to c, then a is preferred to c. In ordered pair notation: if (b,a) € R and

(c,b) € R, then (c,a) £ R or, R is a transitive relation.

The set j/ will always represent a finite collection of objects.

DEFINITION 1.1. A preference relation on J is a binary relation, R (R£ that is irreflexive and transitive.

If R is a preference relation on xf, and (a,b) € R, then

(b,a) <£ R. Otherwise, since R is transitive, both (a,a) and

(b,b) £ R. But R is irreflexive. Preference relations therefore, are antisymmetric, which the previous discussion indicated was a desirable property.

In some applications it is useful to consider binary relations which are irreflexive and antisymmetric. Bogart calls these intransitive preference relations or antisymmetric orderings [6].

Associated with each preference relation on J is a partial order and its associated Hasse diagram. If R is a preference relation on J, for each a,b € xf, define a < b if and only if

(a,b) £ R. Since R is transitive and antisymmetric, ,<) is a partially ordered set. Since R is irreflexive, a / a for all a € xf and hence '<' is what is called a strict partial order on xf.

Example 1.1 shows how a preference relation can be represented in an order theoretic sense by the Hasse diagram of its associated partially ordered set. 6

EXAMPLE 1.1. Let J = {1,2,3,4}, P = {(1,2),(1,3),(1,4),(3,4)},

and Q = {(1,2),(1,3),(1,4)}. P and Q are preference relations on j

In an attempt to quantify the concept of "agreement between preferences", several "distance functions" on the collection of preference relations on J have been proposed and their relative advantages widely discussed [9,23,24].

if will always denote a collection of binary relations. The underlying set will be clear from the context.

DEFINITION 1.2 , A distance function on g is a real valued function, d: £ * £ -> R, satisfying the following properties:

DI: If A,B € £ then d(A,B) 0. Equality holds if and only

if A = B

D2: If A,B € £ then d(A,B) = d(B,A).

D3: If A,B,C €$ then d(A,C) < d(A,B) + d(B,C).

A distance function on £ is usually called a metric and (£,d) a metric space. K. Bogart, generalizing the work of Kemeny and Snell

[18], lists properties DI, D2, and D3, along with four other

"reasonable" properties, as a collection of axioms that any measure of distance on collections of preference relations should satisfy [5]. 7

Before listing these additional axioms I will define some geometrical concepts in collections of binary relations. These concepts are generalizations of Bogart's definitions.

If we consider ~P, the collection of all preference relations on , there is a natural relationship that sometimes occurs among three elements of "P, P, Q, and R. The preferences represented by Q are said to be "between" those of P and R if Q prefers b to a whenever both P and R do and only if at least one does. This relationship is made more precise by the following definition.

DEFINITION 1.3. Let P,Q,R € £. Q is between P and R

(denoted B(P,Q,R)) if and only if PDRCQCPUR. If in addition Q / P and Q / R, then Q is strictly between P and

R (B*(P,Q,R)).

DEFINITION 1.4. Let P,Q £ £. A line segment from P to Q is a sequence, P = P ,P2,...,Pn = Q, of relations from Ç which satisfies the following conditions:

1. If 1 < i < j < k < n, then B*(P. ,P. ,P, ) . — — 1 j K 2. If R £ £ and B(Pi,R,Pi+1) for some i £ (l,2,...,n - 1},

then R=P. or R=P.1. l l+l

Example 1.2 shows that in some collections of binary relations two "points" may determine more than one line. 8

EXAMPLE 1.2. Let J = {1,2,3,4} and £ = ~P. Let P and Q be as in Example 1.1 and R,S € T9 have the following Hasse diagrams:

Then P A S = Q, P U S = R and therefore B*(P,Q,S) and B*(P,R,S).

It is easy to verify that the sequences P,Q,S and P,R,S are both line segments from P to S. Notice that neither P,Q,R,S nor

P,R,Q,S are line segments since R is not between Q and S and

Q is not between R and S. //

Bogart's fourth axiom for a distance function d, on requires that d interact nicely with the betweenness relationship.

The last four axioms are:

D4: Let P,Q,R £ £. B(P,Q,R) if and only if

d(P,R) = d(P,Q) + d(Q,R).

D5: If P = Q {(a,b)}, P' = Q' U { (a,b) }, P / Q, P'/Q',

and P,Q,P',Q'€ £. Then d(P,Q) = d(P',Q').

D6: If {(a,b)}, {(c,d)}, 0 € £, then

d(0,{(a,b)}) = d(0,{(c,d)}).

D7: The minimum positive distance on £ is 1.

Since Bogart was primarily concerned with preference relations, he stated these axioms for the case £ = f3 [5]. Later they were generalized to collections of antisymmetric orderings [6]. 9

If P and Q are as specified in D5, then P,Q is a line segment from P to Q. P',Q' can be viewed as a "translation" of this line segment. Axiom D5 may be interpreted as saying that d is invariant under translations of these types of line segments.

The sixth axiom insures that the distance between two relations will remain unchanged if the labels (elements of xf) are permuted among themselves. The last axiom results in the choice of one for the basic unit of distance.

The following theorem is a combination of Theorems 1 and 2 of

[5]. Let A A B = (AU B)\ (A f} B) and |a| denote the number of elements of the set A.

THEOREM 1.1. Let d: ~P x 1° -> $. d satisfies D1-D7 if and only if d(P,Q) = |P A Q| for all PSQ€7>.

The axiomatic development of distance on the collection of preference relations therefore results in the existence and uniqueness of a distance function on 'fa. This symmetric difference distance function has occurred in many contexts without regard to any axiomatic framework [23,25,34]. Theorem 1.1 provides a characterization of this distance function for collections of preference relations.

Bogart noted in [5] that D3 was not necessary to prove the uniqueness of the distance function on Thus D3 is a consequence of the other axioms. Only weaker forms of DI, D2, and D4 were necessary to obtain the result. These weaker forms are:

D'l: d(P,Q) > 0 for some P,Q € £. 10

D'2: If 0, {(a,b)}€£ then d(0,{(a,b)}) = d({(a,b)},0).

D'4: If P,Q,R € g and B(P,Q,R) then d(P,R) = d(P,Q) + d(Q,R)

(Again, these are stated for an arbitrary collection of binary

relations on

The following is a corollary to the proof of Theorem 1.1.

COROLLARY 1.1. Let d: 7° + R. If d satisfies D'l, D'23

D’4, D5-D73 then d(P3Q) = |P A §| for alt P3Q € 7°. Furthermore3

d satisfies D1-D7.

Bogart does not claim that this reduced list of axioms forms an

independent collection. In [5], he leaves open the problem of

improving this result. The following theorem indicates that by

keeping D2, the fifth axiom may also be deleted from the list.

THEOREM 1.2. Let d: 1° x 1° -> $. If d satisfies D2 and D'43

then d satisfies DS.

Proof: Let P,Q £ 1° with P = Q U{(a,b)}, P / Q. Set

T = {(a,b)}. The following relationships are easily verified:

B(T,P,Q), B(T,0,Q), B(0,T,P), B(0,Q,P).

Using axioms D2 and D'4 and these relationships,

d(P,Q) = d(T,Q) - d(T,P)

= [d(T,0) + d(0,Q)] - [d(0,P) - d(0,T)]

= 2«d(0,T) - [d(0,P) - d(0,Q)]

= 2-d(0,T) - d(P,Q).

Therefore d(P,Q) = d(0,T). 11

If P' ,Q' £ 7° such that P'=Q'UT and P' / Q', then a similar argument shows d(P',Q') = d(0,T). Therefore d(P,Q) = d(P',Q'). //

The following corollary is a result of Theorem 1.2 and

Corollary 1.1.

COROLLARY 1.2. Let d: /° * P ■+ S. If d satisfies D'l,

D2j D'4, D6, and D7 then d(P,Q) = |P A Q] for all P,Q £ ~P. Further­ more d satisfies D1-D7.

Bogart also gives a stronger version of D5 that will be useful in more general contexts:

T5: If P,Q,P’,Q'€tf and P = Q U T, P’ = Q' U T,

q n T•= 0 = Q’ n T, then d(P,Q) = d(P’,Q’).

THEOREM 1.3. Let d: T3 * 1° & and suppose d satisfies D2 and D’4. If P,Q £ 7° such that P = Q U T and Q I) T = 0, then d(P,Q) = i d(0A(afb)}). (a,b)£T Proof: The proof is by induction on the number of elements in T.

If |t| = 1, then P = Q U {(a,b)} and by the proof of Theorem 1.2, d(P,Q) = d(0,{(a,b)}).

Induction Hypothesis: If R,S € X7 such that R = S U K,

S r\ K = 0 and |k| < n (n >_ 2) then d(R,S) = £ d(0,{(x,y)}). (x,y) € K Let P,Q £7° such that P = Q UT, Q OT = 0 and |t| = n.

Let A = {x | (x,y) £ T for some y £ xf} and let a be a minimal element of A (with respect to the partial order induced by P). 12

Let B = {y | (a,y) £ T} and let b be a maximal element of B.

Then (a,b) e T. Notice that if (b,y) € P, then y f? B and if

(x,a) € P, then x A, for any x,y 6 d.

Let R = Q U {(a,b)}. Since R£P, R is irreflexive. If

(x,y) and (y,z) £ R, then (x,z) £ R unless (x,y) = (a,b) or

(y,z) = (a,b). Suppose (x,y) = (a,b). That is, (a,b) and (b,z) £ R.

Since R £ P, (a,z) € P. Since (b,z) € P, z / B and therefore

(a,z) £ T. Since P = Q UT and (a,z) € P, (a,z) £ Q £ R and

R is transitive. Suppose (y,z) = (a,b). That is, (x,a) and

(a,b) £ R. Since R £P, (x,b) £ P and since (x,a) £ P, x A.

Hence (x,b) T. Since P = Q UT and (x,b) £ P, (x,b) £ Q £R and again R is transitive. Therefore R € 'P.

Let K = T\{(a,b)}, then |k| = n - 1 < n, P = R U K,

R K = 0, and by the induction hypothesis, d(P,R) = J d(0,{(x,y)}) (x,y) £ K Since R = Q U {(a,b)} and Q A{(a,b)} = 0, by Theorem 1.2, d(R,Q) = d(0,{(a,b)}). Finally, P P Q = Q £ r c p = p UQ implies

B(P,R,Q). Since d satisfies D'4,

d(P,Q) = d(P,R) + d(R,Q)

= I d(0,{(x,y)}) + d(0,{(a,b)}) (x,y) € K

= I d(0,{(x,y)}) since T=KU{(a,b)} (x,y)€ T and the theorem is true for all n >_ 1. //

COROLLARY 1.3. Let d: 1° x 1° -> %. If d satisfies D2 and I’4, then d satisfies T5. 13

Proof: Let P,Q,P',Q' £7° such that P = Q U T, P' = Q' U T and Qf)T=0 = Q'nT. By Theorem 1.3, d(P,Q) = I d(0,{(x,y)}) = d(P',Q'). // (x,y) £T

The preceding results are directed towards strengthening

Theorem 1.1. To generalize this result to other collections of binary relations, it will be useful to understand how each axiom contributes to Bogart's proof of Theorem 1.1. The uniqueness part of his proof is summarized in three parts.

1. If P,Q € 7° and |P A Q| > 1, then there exists R €

•such that B*(P,R,Q). (A proof of this is included in

the proof of Theorem 1.3.)

2. If P,Q £7°, then there is a line segment in Is,

P = P,,Pn,...,P, , = Q, from P to Q such that 1 2 k+1 P^ A = {(a^,b^)} for i = l,2,...,k. Using axioms k D2 and D'4 (also D5), d(P,Q) = J d(0,{(a.,b.)}). i=l 1 1 3. Axiom D6 then implies all distances are a ¡multiple of

d(0,{(a,b)}) for a,beZ a / b. D'1 and D7 establish

this distance to be one. Since

P A Q = {(a1,bJ),...,(ak,bk)}, d(P,Q) = |P A Q|.

The first part of this proof is a property enjoyed by the collection

7° and is independent of the axioms. Bogart extended the results of

Theorem 1.1 to the larger collection of all antisymmetric orderings on if. The proof is nearly identical, the only difference being that part one of the proof is easier to verify in this larger collection [6]. 14

This property also holds for the collection of all binary relations on that is, 2 (the power set of J x J), and the symmetric difference distance function is the only distance function on this collection satisfying D1-D7. A more interesting problem occurs when smaller collections of binary relations are considered.

In numerical taxonomy, the problem of estimating the evolutionary relationships within a collection of several different species might be approached by considering the relation "is an ancestor of" on the collection of species [14,17,29]. A similar relation, "was copied from" has been considered in the attempt to reconstruct the chronological history of a collection of manuscripts copied directly or indirectly from a common original [11]. In both of these instances, the relations are preference relations; but they also satisfy an additional property which causes them to be "tree posets."

DEFINITION 1.5. A tree poset on J is a preference relation,

P, on jZ which satisfies the following property:

If (a,b) € P and (c,b) € P and a 0 ,c, then either

(a,c) £ P or (c,a) £ P.

The Hasse diagrams of tree posets have a tree-like structure.

The collection of all tree posets on xf will be denoted by x/.

Notice that »ZS/5.

Let f: £ x g -> R and suppose # £ £. If f satisfies any of the axioms D1-D6 on f, then the restriction of f to 'IS will satisfy the same axioms. This general remark implies that the 15

symmetric difference distance function restricted to >7 will satisfy

D1-D6. Since 0 and {(a,b)} € >7 for any a,b £ J, a / b, this distance function also satisfies D7. That this is the only such function is not obviously true. The property in step one of the outline of the proof of Theorem 1.1 does not, in general, hold for the collection of tree posets on xf. This is illustrated in the following example.

EXAMPLE 1.3. Let «

PflQ = { (1,3), (1,4), (2,3), (2,4) } is not a tree poset since neither

(1,2), nor (2,1) 6PHQ. Furthermore, PhQ is the only binary relation strictly between P and Q. Thus there does not exist a tree poset, T, such that B*(P,T,Q), even though |P A Q| = 2. //

The symmetric difference distance function is the only distance function on <7 which satisfies D1-D7. The proof is constructive and follows a series of lemmas.

LEMMA 1.1. Let P € •J such that P / 0. Then there exists

(a,b) € P such that P\{(a3b)} is a tree poset. 16

Proof: Let P € ,/, P / 0. Set B = {y | (x,y) £ P for some

x € J} and let b be a maximal element of B (with respect to the

order induced by P).

Set A = (x | (x,b) £ P} and let a be a maximal element of A.

Then (a,b) € P.

Set T = P\{(a,b)}. T is clearly irreflexive. If (x,y) and

(y,z) £ T then (x,z) £ T unless (x,z) = (a,b). But this cannot

occur since then (a,y) and (y,b) € P, contradicting the maximality

of a in A. Thus T is transitive.

If (x,y) and (z,y) £ T, then either (x,z) or (z,x) € P

since P € J. Therefore (x,z) or (z,x) £ T unless {x,z} = {a,b}.

If {x,z} = {a,b} then (b,y) £ T £ P contradicting the maximality

of b in B. Therefore T is a tree poset. //

LEMMA 1.2. Let P € /. If d: if x «/ -> % satisfies D23 D’43 and 1)6 3 then d(03P) = \p\-d(03{(a3b)}) for a3b £ a / h.

Proof: Let PC«/. By Lemma 1.1, we can construct a sequence of tree posets, P = P^jP^,... ,P^+^ = 0 such that P^ and

P. . = P.\{(a.,b.)}, i = l,2,...,k. Notice that |P| = k. Since l+l i v i i P. n 0 £p. . £ P. U0, B(P.,P. 190) for i = 1,2,...,k. Thus, l i+I i ’ l l+l k by D'4, d(P,0) = £ d(P.,P. The proof of Theorem 1.2 works i=l 1 1+1 equally well for the collection of tree posets. Therefore, since d satisfies D2 and D’4, d satisfies D5, and since P^ = P^+^ U ((a^jb^)} and {(a.,b.)} = 0 U{(a.,b.)}, dfP^P^p = d(0, { (a. ,b.) } for i = 1,2,...,k. Thus, applying axiom D6, if a,b € Z and a / b, then 17

k k d(0,P) = d(P,0) = I d(P.,P.+1) = I d(0,{(a.,b )}) i=l i=l

k = I d(0,{(a,b)}) i=l

= k-d(0,{(a,b)})

= |P|-d(0,{(a,b)}) //

LEMMA 1.3. If d: 40 x ¿7 -> R satisfies D2, D'4, and D6, then

d(P,Q) = \P A Q\*d(0,{(a,b)}) for all P,Q £<3, and any a,b

a 0 b.

Proof: Let a,b € xf, a / b and P,Q € 40. The proof is by

induction on |P H Q|. If P A Q = 0, then since 0 £ 40 and

B(P,0,Q), D2, D'4, and Lemma 1.2 imply

d(P,Q) = d(P,0) + d(0,Q)

= |P|*d(0,{(a,b)}) + |Q|-d(0,{(a,b)})

= |P A Q|-d(0,{(a,b)}).

Induction Hypothesis _1: If R,S £ 0} such that | R fl S | < n

(n > 1) then d(R,S)= |R A S|-d(0,{(a,b)}).

Suppose |P AQ| = n. The proof continues by induction on

|P\Q|. If P\Q = 0, then P £Q. Since B(0,P,Q), we have d(P,Q) = d(0,Q) - d(0,P). Thus by Lemma 1.2,

d(P,Q) = (|Q| - |p|)•d(0,{(a,b)})

= |Q A P|.d(0,{(a,b)}).

Induction Hypothesis 2: If R,S €^7 such that |R OS| = n and | R\S | < r (r >_ 1), then d(R,S) = |R A S| *d(0,{ (a,b) }) . 18

Suppose that [P\Q| = r. By Lemma 1.1, there exists (x,y) £ P

such that P' = P\{(x,y)} is a tree poset. Either (x,y) €PHQ

or (x,y) £ P\Q. If (x,y) £ P A Q, then |P' fl Q| = n - 1 and

¡P* A Q| = |P A Q| + 1. Also, B(P',P,Q) and therefore

d(P,Q) = d(P',Q) - d(P',P).

By induction hypothesis 1, d(P',Q) = |P' A Q|-d(0,{(a,b)}).

Since P = P' U {(x,y)} and d satisfies D5 and D6, d(P,P') = d(0,{(x,y)}) = d(0,{(a,b)}). Therefore d(P,Q) = |P' A Q|-d(0,{(a,b)}) - d(0,{(a,b)}) = |P A Q|-d(0,{(a,b)}).

If (x,y) £ P\Q, then |P’\Q| = r - 1 and |P' O Q] = n. By induction hypothesis 2,

d(P',Q) = |P’ A Q|-d(0,{(a,b)}). Again,

d(P,P') = d(0,{(a,b)}).

In this case, B(P,P',Q) and |P’ A Q| = ¡P A Q| - 1. Hence d(P,Q) = d(P,P') + d(P',Q)

= d(0,{(a,b)}) + |P’ A Q|-d(0,{(a,b)})

= |P A Q|-d(0,{(a,b)}).

Therefore the result holds for all r > 0 and hence for all n >_ 0. //

THEOREM 1.4. Let d: J x ft. If d satisfies D'l, D2, D'4,

D6, andD7, then d(P,Q) = |PA<2| for a It P,QGJ. Furthermore, d satisfies D1-D7.

Proof: Let d: *7 x >»7 R and suppose d satisfies D'l, D2,

D'4, D6, and D7. By Lemma 1.3, all distances are a multiple of d(0,{(a,b)}), which is the same for all a,b £ «f, a / b, by D6.

Axiom D'l implies some distance is positive. Then clearly, 19

d(0,{(a,b)}) is the minimum positive distance, and by D7,

d(0,{(a,b)}) = 1 for all a,b £ Z, a / b. Thus, by Lemma 1.3,

d(P,Q) = I? A Q| for all P,Q £«/. The remarks preceding Example 1.3

show that d satisfies D1-D7. //

The following generalization results from isolating the properties satisfied in the collection «/, necessary for the proof of Theorem 1.4.

THEOREM 1.5. Suppose C has the following three properties:

1. |£| > 2.

2. If P £ S3 then there exists (x3y) €. P such that

P\{ (x3y) } £ £.

3. If P £ If and (a3b) € P, then {(a3b)} £ £.

Then if d: % satisfies D'l3 D23 D'43 D63 and D73 d(P3Q) = |P A Q\ for all P3Q (Lf. Furthermore d satisfies

D1-D7.

Proof: Suppose 5 satisfies 1, 2, and 3 and d: x S -+ R is such that d satisfies the axioms listed in the theorem.

The proof of Theorem 1.2 depends only on axioms D2, D'4 and properties 1, 2 and 3, from which one can easily deduce that 0 £ C.

Thus d satisfies D5 on ¿J. Lemma 1.1 is property 2, Lemmas 1.2 and 1.3 are also true if we add the restriction that {(a,b)} £ £, and the proof of Theorem 1.4 depends only on these lemmas. Finally, d satisfies D1-D6, since these properties are inherited, and d satisfies D7, since 0 and {(a,b)} £ If for some a,b € xf, a / b. // 20

Next we will examine some collections of binary relations occurring

in applied problems which do not satisfy the three properties listed

in Theorem 1.5.

DEFINITION 1.6. Let |^| = n. An n-tree, P, (on J") is a

set of subsets of xf satisfying

1. 0 0. P.

2. j/ £ P and {i} 6 P for all i £ xf.

3. If A,B £ P and A H B / 0, then either A £B or B £A.

An n-tree on is an example of a type of hierarchical classifi­ cation of the elements of xf. Such structures are often the result of clustering algorithms applied to [4,17,29]. In taxonomic applications, this tree structure is useful for classification purposes, the elements of *f representing the various species of a group of contemporary organisms.

An n-tree can be easily represented as a binary relation on the power set of xf in the following way. For each A,B £ xf, define the relation, A < B if and only if BC A (B / A). If P is an n-tree, the collection of ordered pairs, {(A,B) [ A,B € P and A < B} is called the ordered pairs representation of P. The collection of ordered pairs representations of all n-trees will be denoted by xK.

If P then P is a binary relation on the power set of xf and it is easy to show that P is a tree poset.

If P £ Jf then the subset representation can be recovered by setting P' = {)/} U Range(P) where Range(P) = {B | (A,B) € P for 21

some Ac/}, in this case, the ordered pairs representation of

the n-tree P' is P. Both P and P' will be referred to as

n-trees on jf. The intended representation will be made clear from

the context.

The following theorem characterizes those tree posets that are

the ordered pairs representations of n-trees.

THEOREM 1.6. Let U\ =n and. P be a tree poset on the power

set of jf. P is the ordered pairs representation of an n-tree if

and only if P satisfies the following properties:

1. (f, {i}) € P for all i €. xf.

2. If (A3B) €. P3 then A < B and B 0 0.

3. If (J3A) and (j03B) €. P and A f} B / 03 then either

(A3B) £ P3 or (B3A) £ P.

4. If (A3B) £ P3 then (J,B) € P.

Proof: Suppose P' is an n-tree and P = C(A,B) | A < B and

A,B € P' }. Since j/ £ P' and {i} £ P’ for each i £ xf,

(Z{i}) £ P for all ieZ If (A,B) £ P then A < B and B £ P' so B / 0. If W,A),(/,B) £ P and A fi B / 0, then A,B € P' hence A C B or BCA and therefore (A,B) £ P or (B,A) € P.

Finally, if (A,B) € P, then B € P' and B / if, hence J < B and

G/,B) € P. Thus P satisfies 1-4.

Conversely, suppose P is a tree poset and satisfies 1-4. Let

P' = Cxf} U Range(P). Then P', and since (J,{i}) £ P for all i £ Z -Ci} £ P’ for each i £ xf. By property 2, if A £ Range(P) then A / 0, hence 0 P*. Suppose A,B € P' and A H B / 0. If 22

A = Z or B = J then A C B or B c A and P' is a tree poset

If A / xf and B / /, then A,B € Range(P). By property 4

G/,A) and (J,B) £ P and hence, by property 3, either (A,B) or

(B,A) £ P. Thus by property 2, A £ B or B £ A and P' is an

n-tree. //

The restriction of the symmetric difference distance function to

must satisfy D1-D6. If P,Q £ -4'' and P' ,Q' are the corresponding

subset representations and P 0 Q, then there exists A such that

AGP' A Q'. Now 1 < |a| < n since P' and Q' are n-trees.

Therefore there are at least three ordered pairs in P A Q, namely:

(A,{a}), (A,{b}), and G/,A), where {a,b} £ A, a / b. Thus for any P,Q € such that P / Q, |P A Q | >_ 3 and therefore the

restriction of the symmetric difference distance function to /f does not satisfy D7. Notice that D5 and D6 will be satisfied vacuously by any distance function on -V*.

It is possible to define a distance function on that satisfies

D1-D7. Let d': R such that d'(P,Q) = y |P A Q| for all

P,Q €X Since d' is a scalar multiple of the symmetric difference

The n-trees

on / = {l,2,...,n} have |P A Q| = 3 and therefore d'(P,Q) = 1. 23

This is the minimum positive value on and therefore d' satisfies

D7. Example 1.4 shows that this is not the only distance function on

Jf which satisfies D1-D7.

EXAMPLE 1.4. Let xf = {l,2,...,n}. For any binary relation, P, on the power set of J, define P° = {(A,B) €P | A//}. It is straightforward to verify by set theoretic arguments that if P, Q, and R are binary relations on the power set of xf, then

1 . (P A Q)° = p° A Q°

2 . (P U Q)° = P° U Q°

3 . PPQCR5PUQ if and only if

| P A Q | = | P A R| + | R A Q| [25] .

Define d": r by d"(P,Q) = | |P° A Q° | for all

P,Q £ Jf. If P,Q€# and P / Q, then there exists A c xf and a,b £ A (a / b) such that (A,{a}), (A,{b}), and C/,A) are elements of P A Q. Thus |P° A Q° j >_ 2. The n-trees P and Q defined in the text preceding this example have |P° A Q°| = 2. Thus d" satisfies D7. Axioms D1-D4 are verified using 1, 2, and 3 above, and D5 and D6 are satisfied vacuously. Finally, let R € 00 have

Hasse diagram: {1} {2}{3} {4}>>>{n}

R: {1,2,3} 24

Then |P A R| = 7, |P° A R°| = 5 and therefore d’(P,R) = y , d"(P,R) = | .

Thus d' and d" are two distinct distance functions on which satisfy D1-D7. //

This example indicates the importance of axiom D6 to the uniqueness of the distance function in the case of tree posets or preference relations. In order to obtain uniqueness results for arbitrary collections of binary relations, it will be necessary to have an axiom that is applicable to all collections £, but preferably one consistent with D6 in case f = f3 or & = J .

DEFINITION 1.7. Let P,Q £ S, P / Q. P and Q are

£-adjacent if and only if for each R £ Z x Z, B*(P,R,Q) implies

R <£ £.

Bogart showed that if P,Q 6 T5 and |P A Q, >_ 2 then there exists R £ f3 such that B*(P,R,Q) [5]. Thus /’-adjacency is equiva­ lent to |P A Q| = 1. For preference relations, the combination of axioms D5 and D6 forces the distance between all /-adjacent relations to be the same.

EXAMPLE 1.5. Let xf = {1,2,3} and P,Q,R 6 Z with Hasse diagrams: 25

Then P fl Q = {(1,3),(2,3)} *7. It is easy to check that P and

Q are*/-adjacent. Since |P A R| = 1, P and R are also

^-adjacent. But d(P,Q) = ¡P A Q| = 2 and d(P,R) = 1. Since

this is the only distance function on «7 satisfying D1-D7, it is

impossible to have a distance function on t7 which measures distances

between all >7-adjacent relations the same, and at the same time

satisfies D1-D7. The next example explicitly demonstrates this

impossibility. //

EXAMPLE 1.6. Let xf={l,2,3,4) and T15T2,...,T have Hasse

diagrc----

4 2

<4

T4:

<►3

ll

It is easily verified that B(TpT2,Tg), B(T2,T3,T<.), and

BfTpT^jT,.). >7-adjacencies are indicated by line segments in the

following diagram. T^,T2,T^,T^ and T^,T^,T^ are both line

segments in Z 26

If f: *7 x fe7 -> R and f satisfies D'4, then

f(TrT5) = f(T1}T2) + f(T2,T3) + f(T3,T5) and

f(T15T5) = f(TpT4) + f(T4,T5). Although all of these are distances between -7-adjacent relations, they cannot all be the same. Notice that |T A T5I =2. //

If P and Q £ £ (P / Q), then it is possible to construct a

line segment from P to Q in <£. This line segment is a key to determining the distance between P and Q. The following lemmas demonstrate the construction of a line segment from P to Q.

LEMMA 1.4. Let P,Q £ $a nd # = {7? € £ | B(P,R,Q)}. If

R,S £ 73 and A £ £ such that B(R,A,S), then A £ 73.

Proof: Let R,S £ 73, h£ £ such that B(R,A,S). Then

PAQCRCPUQ and P 6 Q £ S c p UQ, Therefore, since B(R,A,S),

P 6 Q £ R H S £ A £ R U S £ P u Q and hence B(P,A,Q) . Thus A € 73./ /

LEMMA 1.5. Let P,Q € g sueh that P 0 Q. Then there exists a sequence, P = P^P^,... ,P^+1 = Q (k >_1), of elements of $ satisfying the following properties:

1. P^ is £-adjacent to = l,2,...,k.

2. B^(PiSPi+1,Q) for i = l,2,...,k - 1 (if k>l).

Proof: If P and Q are ^-adjacent then P,Q is the required sequence. If P and Q are not ¿f-adjacent, then

= {R £ £ | B*(P1,R,Q)} / 0. Lemma 1.4 implies that for any

S € , {R £ £ | B*(P1,R,S)} £ Thus there exists P2 £ such 27

that P2 is £-adjacent to P . Let P2 be the second element of the sequence.

Suppose the ith element of the sequence, P^, has been chosen

(i >_ 2). Set # = {R £ £ | B*(Pi,R,Q)}. If JSL = 0, then P£ is ...,P^,Q.

If 73. 0 0, choose P. , £ 73. such that P. , is

= 0. The desired sequence then is P = P^,P2,...,P^,P^+^ = Q.

P. is ^-adjacent to P. . by choice, and, since P. n £ 7?

B*(P^,P^+1,Q) for each i. //

In view of Lemma 1.4, the sequence in Lemma 1.5 is a line segment from P to Q in £.

LEMMA 1.6. If P = F23’'’iFk+l = ® +s a sequence of elements of G satisfying B^(P^3P^+^3Q) for i = l323 ... 3k - 13 then

(P. A P. Pl (P. A P.,.,) = 0 for all i3j such that 1 < i < j < k. i i+l 3 3+1 ~ ~ Proof: Fix i and j such that 1 <_ i < j £ k. If

H, £ {i,i + l,...,j}, B*(prp£+1’Q) and therefore P. n Q £ p.+i nQ£...cp. OQC p_. + i cp. u Q cp.^ u QC ... c p. UQ

Let x £ (P^ A pi + 1) • Then either x £ Pj^P^+j or x £ Pi+lXPi-

If x £ PAP. ., then since P. nQ£P.., x £ Q. Hence i i+l l i+l x

If x € P. AP. , then, since P. . Q P. U Q, x € Q. But i+l i’ ’ i+l l 28

P^ + 2 ft Q £= Pj ft Q £ Pj + l’ hence x £ Pj Pj+l’ and therefore x

LEMMA 1.7. If P = P^P^}...jP^ = Q is a sequence of elements of £ such that B3:(P^3P^^}Q) for i = 1,2,__ ,k - 1, then k P A Q = U (Pj A P^+^- Furthermore, this union is pairwise disjoint. i=l Proof: Let 73 = {R £ £ | B(P,R,Q)}. Then P £ '8. If

P^ £ ~Z3 (i >_ 1), then, since B(P^,P^+^,Q), Lemma 1.4 implies

P. , £ 73. Thus P. £ 73 for j = 1,2,... ,k + 1. Hence i+l j J PAQcp. c p UQ for j=l,2,...,k+l.

Suppose x < P A Q. Then either x £ P n Q, or x £ P H Q.

If x £ pn Q, then x £ P. H P. , for each i and therefore i i+l k x £ (J (P- A P- J • If x O U Q then x P. U P. for each i r rf i i+l i i+l 1-1 k k and therefore x 4. (J (P. A P. A. Thus kJ (P. A P. A £ P A Q. r , l i+l . , l i+l i=l i=l Let x £ P A Q. Then either x £ P\Q or x € Q\P. Suppose x £ P\Q and let j be the smallest integer such that x f. P., (1 < j k + 1) k J Then x € P. . and therefore x £ P._. A P. £ U(p-; A P, A. If J - J J i=l i 1 x £ Q\P and j is the smallest integer such that x £ Pj fl < i < k + 1), then x £ PAP. .. Thus x £ P. . A P. and there- — 13-1 3-1 3 k fore x £ \J (P. A P. A . . , i i+l i=l 29

k Hence P A Q = {J (P^ A • By Lemma 1.6, this union is i=l pairwise disjoint. //

Let d: £ x £ -> R and suppose d satisfies D'4. If P and

Q £ £, then d(P,Q) is completely determined by the distances between the £-adjacent relations composing any line segment from P to Q. Lemma 1.5 guarantees the existence of such a line segment in any collection £. In the case of preference relations,

/’-adjacent relations have symmetric difference one and the additional axioms were sufficient to determine these distances. If

T and S are tree posets and «/-adjacent, the additional axioms determined that d(T,S) = |T A S|. According to Lemma 1.7, this would be desirable in the general case. For along any line segment from P to Q in £, regardless of the number of relations composing it, the elements of P AQ are exactly those in the symmetric differences of the ^-adjacent relations in that line segment.

DEFINITION 1.8. Let P,Q £ C be such that P and Q

£-adjacent. The degree of adjacency of P and Q in S is

DA(P,Q) = |P A Q|.

The degree of adjacency of a ^-adjacent pair is a measure of the differences between the two relations. The preceding discussion is the basis for the following axiom which will replace DI, D6, and

D7 in the determination of a distance function on £. 30

Cl: If P,Q £ £ are i?-adjacent, then d(P,Q) = DA(P,Q) .

If P = Q, then d(P,Q) = 0.

THEOREM 1.7. Let d: £ * £ •+ S. If d satisfies D’4 and Cl

then d(P,Q) = ¡P A for all P,Q £ C. Furthermore, d satisfies

D1-D6.

Proof: Let P,Q € £. If P = Q then by Cl, d(P,Q) = 0 = |P A Q|

If P and Q are ¿^-adjacent, then d(P,Q) = DA(P,Q) = |P A Q|. If

P/Q and P and Q are not ^-adjacent, then by Lemma 1.5, there exists a sequence, P = P ,P , ...,P, = Q (k >_ 2)} of elements of £ 1 Z K + j. such that P. is ^-adjacent to P. , and B*(P.,P. .,Q). Using l J i+l i i+l D'4, Cl, and Lemma 1.7,

k d(P,Q) = I i=l

-- 1T=1aa-la

= .1 |P± 4 Pi+1l 1=1

’ I A Cpi 4 I

1=1

= |P A Ql.

Since d is the restriction to £ of the symmetric difference distance function on the collection of all binary relations on -Z, d satisfies

D1-D6 // 31

Theorem 1.7 characterizes the symmetric difference distance

function on by a pair of axioms. Since D5 is satisfied

vacuously by d on Jf, the relationship between T5 and this

collection should be considered. This relationship is determined,

in general, by the following corollary to Theorem 1.7.

COROLLARY 1.4. Let d: + R. If d satisfies D’4 and

Cl, then d satisfies T5.

Proof: By Theorem 1.7, d(P,Q) = |P A Q| for all P,Q £

If P,Q,P',Q' £ £ and P = Q U H, P' = Q' U H, and

Q A H = 0 = Q' H H, then PAQ = H=P'AQ'. Therefore,

d(P,Q) = |H| = d(P’,Q’). //

The following examples prove the independence of axioms D'4

and Cl.

EXAMPLE 1.7. Let J = {1,2,... ,n} and R,R’ £

diagrams: 1 n 2 n-1 3 3' 2 n-^-1 1 a n

Since 0 €x7 and B*(R,0,R'), R and R' are not 47-adjacent. Define p: «7 x «7 R by f~|P A Q|; if {P,Q} / {R,R*}

P(P,Q) = < [ |P A Q| - 1; otherwise 32

Clearly p satisfies Cl (and D1-D3) but

p(R,R') = ]R A R'| - 1 / |R A R'i = |R| + |R'| = p(R,0) + p(0,R').

Since B(R,0,R') and p(R,R') / p(R,0) + p(0,R'), P does not

satisfy D'4. //

EXAMPLE 1.8. Let Z = (1,2,. . . ,n}. Define p' :

by p'(P,Q) = 2|P A Q|. The distance function p' satisfies D1-D4

but not Cl. //

Another type of discrete structure often considered in the

biological sciences is called a "tree of subsets", or "cladistic

character" on Z [14].

DEFINITION 1.9. Let P be a collection of non-empty subsets

of . P is a tree of subsets of jf (or cladistic character on

if and only if / 6 P and for each A,B £ P, A B / 0 implies

A £ b or B S a.

A tree of subsets of xf may be represented as a binary relation

on the power set of xf in the same way as n-trees. In certain biological applications, the resulting Hasse diagram may represent estimates of ancestral relationships among the various species represented by elements of xf. These estimates usually result from the consideration of certain biological characteristics which distinguish some of the species from others.

Let g be the set of all ordered pairs representations of trees of subsets of xf. Since is a collection of binary relations (on 33

the power set of J), Theorem 1.7 again establishes a unique distance function on £ that satisfies D1-D6 and Cl.

In some applications, a "normalized" form of the symmetric difference distance function is desired [4,5,32]. That is, the range should be contained in the interval [0,1] and the values of 0 and

1 should be attainable. If the maximum value of d on any collection is M, then d may be normalized by dividing all distances by M. Thus

dN(P,Q) = i d(P,Q) = lP- £or P,Q € £ is a normalized form of the symmetric difference distance function on If • Maximum values for d on and -V are given in the following theorems.

THEOREM 1.8. Let = {1323...3n}. Then

max {\P L Q\} = n2 - n. P3Qffi

Proof: If P £ 1°, the maximum number of ordered pairs in P 1 2 is _ = —Z—- since P is irreflexive and antisymmetric. Thus, V J f 2 - 1 if a = max | P A Q | , then a <_ 2 -—= n2 - n. p,Q£p I L 1 Let P = {(i,j) j i < j, i,j £ xf}. Then P £7° and |p| = f" •

If P = {(j,i) I (i,j) € P}, then ]P A P’] = 2• ” = n2 - n and therefore a = n2 - n. //

COROLLARY 1.5. If J = {13 23 . .. 3n}, then max {|P A Q|} = n2 - n P,QGJ Proof: Since *7 Q. p, if a = max {|P A Q|}. Then P,Q£s7 a £ max {|P A Q|} = n2 - n. P,QeP 34

Since P and P', defined in the proof of Theorem 1.8 are tree posets, a >_ n2 - n. Thus a = n2 - n. //

LEMMA 1.8. Let T' be an n-tree on {l,2,...,n}. If

T' = {4^, .. . ,A , {1}, {2}, . .., {nj}, then r <_n - 1.

Proof: Let G = (V,E) be the graph with V = T' and {A,B} £ E if and only if AflB/0, A,B e T' and for each X e T' such that

A c X c B, or B £X £A, either A = X or X = B. (This graph is isomorphic to the Hasse diagram of the ordered pairs representation

T, of T'.) The graph G is easily shown to be a tree and there­ fore |e| + 1 = |v| [7]. Each of the vertices may be identified with at least two different edges in E, namely

(A. ,xb and {A. ,xb where x\x! CA.. Thus |e| > 2r and

|v| = |E | + 1 2r + 1. Since |v| = n + r, n + r >_ 2r + 1 and therefore n - 1 >_ r. //

THEOREM 1.9. Let = {1,2,...,nj and T’ be an n-tree on J.

Let T be the ordered pairs representation of T'. Then jyjJ <_ 7-22 - 7-2.

Proof: If n = 2 then T' = {{1},{2},{1,2}} and

2 = [t| = 22 - 2. Assume that if S' is a k-tree (on {l,2,...,k}) and k < n, then |s| < k2 - k where S is the ordered pairs representation of S'. Suppose T' is an n-tree (n >_ 3) and let

A1}A2,...,Ar be the nodes on the Hasse diagram directly above V.

That is, {A^,^} is an edge of the graph of T' as defined in

Lemma 1.8. For i=l,2,...,r, let r = (A € T' | A <= A±}. Under 35

an appropriate relabeling of the elements of A^, each T| is an

|A^|-tree on {1,2,...,|A^|}. By the induction hypothesis, we

have It.I < |A.12 - |A.| where T. is the ordered pairs

representation of T|. T has the ordered pairs in K (for each i)

plus the pairs of the form G/,A), A £ T|. By Lemma 1.8,

|T|| £ 21 A± I " !• Hence,

|T| = Ï |T.| + I |T!| i=l 1 i=l 1

£ i C|A I2 - |AJ) + I (2|A | - 1) i=l i=l

r = 7|A.|2-n+2n-r i=l 1

= I |AJ2 + n - r. i=l

A simple induction argument verifies that if a^ + a^ +...+ ay = n,

a^ >_ 1 and 1 < r £ n, then a2 + a2 +. . .+ a2 <_ n2 - 2n + r. Thus,

|t| <_ (n2 -2n + r)+n-r=n2-n. //

COROLLARY 1.6. If J = {1323... 3n}3 then

max {|P A §|} = 2nf - 4n. P,Q£^ Proof: Since {J,{1},{2},...,{n}} £ P' 0 Q' for any n-trees

P' and Q', ¡P A Q| >_ n for all P,Q € ^V. Let a = max {¡P A Q|}. Then by Theorem 1.9 p,Q£>r a £ 2 (n2 - n) - 2n = 2n2 - 4n

Let A, = {1,2, ...,k + l} and B =xf\Ak, k=l,2,... ,n - 1. Then 36

P' = {A ,A2,...,A ,{l},...,{n}} and

Q' = {B ,B2,...,Bnl,{l},...,{n}} are n-trees on J. If P and Q are their ordered pairs representations, then |P| = |q| =n2 -n,

|p H Q| = n, and therefore |P A Q| = 2(n2 - n) - 2n. Thus a = 2n2 - 4n. // CHAPTER 2

DISTANCES BETWEEN PARTITIONS AND OTHER SETS OF SUBSETS

The discrete structures studied in Chapter 1 often have more

than one common representation. Preference relations were

represented as binary relations, strict partial orders, and Hasse

diagrams or directed graphs. N-trees were defined as sets of

subsets, then represented as binary relations. Example 1.4 gave

a second representation of n-trees as binary relations and showed

the dependence of the measure of distances upon the particular

representation used. The relative merits of one representation over

another must be decided in light of the applications under considera­

tion. The following example examines the interplay of our intuition

about distance concepts and the choice of representation.

Collections of sets of non-empty subsets of J will be denoted by £'. Jf' and S' are the collections of n-trees and trees of

subsets of «f, respectively. and £, defined in Chapter 1,

are their ordered pairs representations.

EXAMPLE 2.1. Let = {a,b,c,d} be a collection of four species or types of organisms. Estimates of the evolutionary history of the organisms in j/ may be represented by trees of subsets of xf. Let

P',R' € S' have the following Hasse diagrams, where P and R are the associated elements of £.

-37- 38

In both trees, the node labeled -4, represents the most recent

common ancestor of a, b, c, and d. In P, the node {a,b}

represents a descendant of xf which is the most recent common

ancestor of a and b. The structure of R suggests the existence

of a descendant of Z which is also the most recent common ancestor of a, b, and c. Since both of these hypothetical organisms may have

existed, we may want to combine the information and form a new tree of subsets, S.

The axiomatic development of distances between trees of subsets is dependent upon the concepts of adjacency and betweenness. The representation of evolutionary histories should yield adjacency and betweenness relations consistent with our intuition, if we are to accept the resulting distance function(s). 39

Considering the ordered pairs representations, there is exactly

one element of £, strictly between P and R, namely Q = P fl R.

Notice that ({a,b,c),{a,b}) £ S but neither P nor R. Even if our intuition leads us to consider S to be "between" P and R, the previous definition of betweenness does not allow it. In the representation as sets of subsets, however,

P' A R' S S' S P' U R' and P' D R' £ Q' £ P' U r« .

This is the basis for a betweenness relationship in the context of sets of subsets. //

DEFINITION 2.1. Let P,Q,R £ £'. Q is between P and R if and only if P D R £ Q £ P U R. Q is strictly between P and R if B(P,Q,R) and P / Q and Q / R. (Since only the elements of the structures in £' have changed, the same notation is used in this context as in Definition 1.3.) [cf. 25]

Axioms D1-D4 and D7 remain as desirable properties for a distance function on C to satisfy. The fifth and sixth axioms must be translated to the language of sets of subsets.

S5: If P,Q,P',Q' € £' are such that P=QU {A},

P' = Q' U (A) (A£/) and Q n (A) = 0 = Q' fl {A}, then

d(P,Q) = d(P',Q'). 40

E6: If {/},{»/,A}, {/,B} £ £', then

d({/},{J,A}) = d({/},W,B}).

The choice of {J} in E6 to be the "minimal" element of £'

is because of the desire to apply these axioms to the collection

The following lemmas lead to a result about the existence and uniqueness

of a distance function on .

LEMMA 2.1. Let P,Q £ £’, then P A Q € S’ and B(P,Pr\Q3Q).

Proof: Let P,Q € <£' . Then clearly 0 0 P A Q and £ P A Q.

If A,B £ P A Q and A A B = 0, then A,B € P £ £' and hence

A £ B or BSA. Thus P A Q £ £'. Clearly B(P,PAQ,Q). //

LEMMA 2.2. Let P,Q £ S’ and suppose P £ Q such that

|P A Q\ > 1. Then there exists R £ S’ such that B*(P,R,Q).

Proof: Let A £ Q\P and set R= PU {A}. Since |p A Q| >1,

R / Q and since A 0- P,R / P. Clearly 0 £ R, 0 0 R and if

A,B € R, then A,B £ P. Hence A A B / 0 implies A SB or B£A.

Thus R € S' • Finally, since P A Q = P£R and R S Q = P UQ, we have B*(P,R,Q). //

LEMMA 2.3. Let P,Q € S’. If P 0 Q, then there exists a sequence, P = P^P^,... ,P^+1 = Q, of elements of S’ which satisfies B(P^,P^+^,Q) and \P^ A Pi+1 1 for i = 1,2,...,k. k Furthermore P A Q = (J (P. A P...) and this union is pairwise • I' I'T’ J Z=1 disjoint. 41

Proof: In view of Lemmas 2.1 and 2.2, the proof of Lemma 2.3

is similar to that of Lemmas 1.5, 1.6, and 1.7. //

LEMMA 2.4. Let P,Q € Z’ be such that (P A = 2. If

d: £' S satisfies S5, then d(P,Q) = d({J},U} U (P A Q)).

Proof: Without loss of generality, P = Q (J {A}. Clearly

U),W,A} € £' . Thus by S5, d(P,Q) = d({Z},{xT,A}) = d({/},U} U (P A Q)). //

LEMMA 2.5. If d: £' * £' ->■ E satisfies D2 and D’4, then d satisfies S5.

Proof: The proof is the same as that of Theorem 1.2 with the appropriate changes of notation. //

THEOREM 2.1. Let d: £’*%'+ S. If d satisfies DI, D2, D’4,

E6, and D7, then d(P,Q) = (P A Q\ for all P,Q £ £'. Furthermore d also satisfies D3, D4, and S5.

Proof: Let P,Q £ . If P = Q, then by DI, d(P,Q) = 0 = |P A Q|. If P / Q then by Lemma 2.3, there exists a sequence, P = PiiP2’‘•,Pk+l = in satisfying

B(pi,pi+1,Q) and |pi A Pi+1l = 1 for i = l,2,...,k. By Lemmas

2.4 and 2.5, d(P.,P.+1) = d({^},{xf,A.}) where {A.} = P^P^.

Since d satisfies E6, all these distances are the same. Using D'4, k d(P,Q) = y d(Pi,pi+1) = k*d({w?},{J’,A}) for any A £ J, (A / /, i=l A / 0) . Thus DI and D7 imply d(P,Q) = k. By Lemma 2.3,

P A Q = {Aj,...^}, hence d(P,Q) = k = |P A Q] . That d satisfies the additional axioms is easily verified. // 42

Since Jf' £ £' and there are pairs of n-trees with symmetric

difference one, the restriction of the symmetric difference distance

function to Jf' is a perfectly good distance function which

satisfies D1-D4, S5, E6, and D7. Axiom E6 is satisfied vacuously

and the uniqueness of this distance function is no longer clear.

A slight modification of E6, however, guarantees uniqueness.

N6: Let $ denote the trivial n-tree. That is,

$ = W} U {{a} | a €/}. If A,B £ Z, A,B O,

then d($,$ U {A}) = d($,$ U {B}).

THEOREM 2.2. Let d: Jf' x Jf' -+ S. If d satisfies DI, D2,

D’4, N6, D7, then d(P,Q) = |P A £?[ for all P,Q € Jf'. Furthermore d satisfies DZ, D4, and S5.

Proof: The proof is similar to that of Theorem 2.1. Notice that Lemmas 2.1, 2.2, 2.3, and 2.5 remain true for n-trees. //

For each new collection of sets of subsets, the sixth axiom must be examined to see if it applies. If it is satisfied vacuously, there is little hope of obtaining results similar to those in

Theorems 2.1 and 2.2. In Chapter 1, we considered axiom Cl as an alternative to this constant revision. This axiom is also appropriate for collections of sets of subsets, ¿^'-adjacent is defined similarly to g-adjacent and the definition of degree of adjacency remains the same. The proofs of Theorem 1.7 and its supporting lemmas are set- theoretic in nature and remain valid for any collection £'. The proof of Theorem 2.3 is essentially a translation of that of Theorem

1.7, and therefore it is omitted. 43

THEOREM 2.3. Let d: S' x S' ->■ R. If d satisfies D'4 and

Cl, then d(P,Q) = J.P A <2| for all P,Q £ S'. Furthermore, d also satisfies D1-D4 and S5.

Trees of subsets of xf, and n-trees on xf, are examples of hierarchical classifications of the elements of J. Partitions on xf are often the result of non-hierarchical classification procedures. They frequently arise in behavioral science experiments

[8,12].

DEFINITION 2.2. Let P be a collection of subsets of xf.

P is a partition of J if and only if (_) A = xf and A,B £ P, A£ P (A / B) implies A A B = 0, and A,B / 0.

The collection of all partitions of J will be denoted by

It is natural to apply Theorem 2.3 to this collection to obtain a distance function. This particular distance function exhibits some undesirable properties when considered in the context of the usual applications.

EXAMPLE 2.2. Let Q = {{1},{2} ,{3},{4}} and R = {{1,2,3,4}}.

Q and R are partitions of 0 = {1,2,3,4}. Suppose P is a partition of xf and P is between Q and R (in the sense of

Definition 2.1). Then RAQ=0cpCRUQ. If {i} £ P for some i £ J, then necessarily, P = Q. Otherwise, P = R. Thus

R and Q are x9'-adjacent, despite the fact that they seem to be as far apart as possible. Let P = {{1,2},{3,4)}. The groupings 44

represented in P are "stronger" than those in Q, and "weaker" than those in R. Our intuition leads us to consider a different type of "betweenness" relation to use in the context of partitions.

Notice also that neither R £ Q nor Q £ R. The relation of set inclusion is inappropriate for capturing this concept of "stronger" or "weaker".

DEFINITION 2.3. Let P,Q €/£>’. P is a refinement of Q, denoted P < Q, if and only if for each A € P, A £ B for some

B € Q. If R € &' and R < P and R < Q, then R is a common refinement of P and Q. If R is a common refinement of P and

Q and for every S 6 £)', S < P and S < Q implies S < R, then

R is called the meet of P and Q, denoted R = P A Q.

It is a standard result that for all P,Q €x9', the meet of P and Q exists and is given by PaQ={a|A=BHC for B £ P,

C € Q, B fl C / 0}. "Meet" is a lattice theoretic term. It is well- known that the collection of partitions of d form a lattice.

DEFINITION 2.4. Let P,Q,R£>(9'. Q is p-between P and R

(denoted Bp(P,Q,R)) if and only if P A R < Q and for each A £ Q,

A £ B for some B € P u R. Q is strictly p-between (B*(P,Q,R)) if Bp(P,Q,R) and P / Q and Q / R. P and R are Q’-p-adjacent if for each Q € x9', Bp(P,Q,R) implies P = Q or Q = R.

EXAMPLE 2.3. Let / = {a,b,c,d,e,f} represent six prisoners who are to be reassigned cells in a prison. Two guards are asked to 45

group the prisoners in a way that will minimize potential conflicts

within each cell, and at the same time keep the number of cells to a minimum. Consider the following groups (partitions) of *f.

P = {{a,b,c},(d,e},(f}}

Q = {{e,a),{d,b,c),{f}}

These groupings agree on the fact that b and c can get along well, and that f should have his own cell. The partition P A Q represents this agreement.

P A Q = {{a),{b,c},{d),{e},{f}}.

Unfortunately, the assignment suggested by P a Q requires five cells. A "compromise" assignment could be chosen from among the partitions that are p-between P and Q. Other than P, Q and P A Q, these are:

{{a,b,c),{d),{e},{f}}

{{a},{b,c),{d,e},{f}}

{{a,e),{b,c),{d},{f}} //

There is another operation associated with the lattice of partitions called the "join." This corresponds to set union in the same way that "meet" corresponds to set intersection.

DEFINITION 2.5. Let P,Q £ . If S £ &' and P < S and

Q < S, and for every T £ /)', P < T and Q < T implies S < T, then S is called the join of P and Q, denoted S = P v Q.

The fact that >©' is a lattice follows from the existence of joins and meets for any pair of partitions in x9'. Another possibility 46

for the definition of p-between is to require PaR

In most applications, the join operation yields a partition that is too "strong" to be considered "between" P and R. In Example 2.3,

P VQ = {{a,b,c,d,e},{f}}. If P V Q were "between" P and Q, then this assignment should be very desirable since only two cells are required. Notice that this assignment would place a and d in the same cell although both P and Q agree that they should be separated.

E. L. Perry formulated the following concept of betweenness for partitions [22]. It turns out that it is equivalent to Definition 2.4.

DEFINITION 2.6. Let P,Re£>’. If {x,y}^/, x/y and either {x,y} £ A fl B for some A £ P, B £ R, or {x,y} <•£ A for every A 6 P U R, then P and R agree on If Q € £)' and Q agrees with P and R on {x,y} whenever P and R agree on {x,y}, for all x,y e J, then Q is between (Perry)

P and R.

Theorem 2.4 verifies the equivalence of Definitions 2.6 and 2.4.

THEOREM 2.4. Let P,Q,R £&'. Then Bp(P,Q,R) if and only if

Q agrees with P and R on {x,y} whenever P and R agree on for all x,y e J, x / y.

Proof: Suppose Bp(P,Q,R) and P and R agree on {x,y}.

If {x,y} £ A fi B for some A € P, B £ R, then since

AD B £ P a R < Q, there exists C £ Q such that {x,y} £. C. Hence

Q agrees with P and R on {x,y}. 47

If {x,y} £ c for some C £ Q, then, since Bp(P,Q,R),

{x,y} £ A for some A € PUR. Thus {x,y} $7 A for all A £ P U R implies {x,y) g C for any C £ Q and hence Q agrees with P and R on {x,y}.

Conversely, suppose Q is between P and R in the sense of

Perry. Let C £ P a R. Then C = A A B for some A £ P, B € R.

If x £ C, then x £ C' for some C'€Q. If {x) = C, then clearly CSC'. If y £ C, y / x, then {x,y} SAAB. Since

P and R agree on {x,y}, Q and P also agree, hence

{x,y} £ A A C' £ C' . Thus CSC' and therefore P A R < Q. If

C £ Q and z £ C then there exists A € P, B £ R such that z € A and z € B. If w e and w 0 A U B, then P and R agree on {z,w}, thus Q agrees with P and R, hence w 0 C.

Hence C £ A U B. Suppose C A and C 5= B. Then there exists

{x,y) £ C such that x 0 A and y 0 B, x / y. Since C £ A l> B, y £ A and x € B. Thus P and R agree on {x,y} and therefore

{x,y} C' for any C' £ Q, a contradiction. Thus either C SA or C S B and therefore Bp(P,Q,R). //

Let d: <0' x Q' R. The following axiom is the analog of D4.

P4: Let P,Q,R £ . Bp(P,Q,R) if and only if

d(P,R) = d(P,Q) + d(Q,R).

(P'4 is stated similarly.)

Perry [22] defined a distance function on which satisfies a collection of axioms, one of which is P4. With the correct 48

representation of partitions as binary relations, Perry's distance function is nothing more than the symmetric difference distance function on the associated binary relations. The representation begins with an arbitrary but fixed total order '<', on Z For any partition P, associate the binary relation P . R

PR = Ua,b) | {a,b} £ A, A € P, a < b, a / b).

Notice that PD = Qn if and only if P = Q for all P,Q € £>' . R R

EXAMPLE 2.4. Let / = {1,2,...,8} with the usual integer order. P = {{1,2,3),{4,6),{5},{7,8}} and Q = {{1,3,5,7),{2,4),{6,8}} are partitions of Z The Hasse diagrams for Pp and Qp are: ,3 I 7

" 5 P 2 6 R 5 t 8 Qr: 4 f 8 3 1 4 4 7 2 ¿6 ■ < 1

Notice that the cells of the partitions correspond to the totally ordered subsets ("connected components") of the Hasse diagrams.

Also P and Q agree on {x,y} if and only if x and y are in the same component in PD and QD or if they are in different components in both Pp and Qp.

THEOREM 2.5. Let P3Q3R € . Bp(P3Q3R) if and only if

B (i’pi Qpi Rp) .

Proof: Suppose Bp(P,Q,R) and let (a,b) £ Pp fl Rp. Then there exists A £ P, B £ R such that {a,b} £ A n B. Hence there 49

exists C £ Q such that (a,b) £ C and therefore (a,b) € QR.

Thus P n RD £ Q . Let (a,b) € Q then (a,b) £ C for some R K R K C £ Q, and C £ A for some A £ P u R. Hence (a,b) £ PR or

(a,b) £ Rr and hence, BCP^Q^R^.

Conversely, suppose B(Pn,QD,Rn) and let x,y € J, x < y R R R such that P and R agree on (x,y). If there exists A £ P,

B £ R such that (x,y) £ A fi B, then (x,y) £ PR A RR £ Qr and hence (x,y) £C for some C £ Q. Thus Q agrees with P and R on (x,y). If (x,y) A for any A £ P u R, then (x,y) / PR U RR, hence (x,y) {7 Q and therefore (x,y) $£ C for any C £ Q. Again

Q agrees with P and R. By Theorem 2.4, Bp(P,Q,R). //

Perry's distance function is defined as follows:

dp(P,Q) = y 6(x,y) for all P,Q£ £) x,ye*f x/y

if P and Q agree on (x,y) where 6(x,y) otherwise for all x,y £ J, x / y.

EXAMPLE 2.5. Let Z,P,Q be as in Example 2.4. Then

dp(P,Q) = 6(1,2) + 6(1,3) +...+ 6(7,8)

=1 +0 +...+ 1

11

Notice that in this case, |PR A QR| = 11. 50

-T-H--E-O--R-E--M-- --2--.6--. Let P,Q £ 6' with Pyy„ and Qyy„ the associated binary relations cn J. Then dp(P,Q) = \P^ A Q|.

Proof: The following statements are equivalent:

1. (a,b) £ (PR A Qr)

2. (a,b) € PR\QR or (a»b) e QrXPr

3. P and Q do not agree on {a,b}

4.

Thus |PR A Qr| = £

Perry characterized his distance function on by a collection of four axioms. When these are translated to the binary relation representation, they correspond to D1-D4 and a combination of D5 and

Cl.

The following axioms are analogs of D5 and D6. Let d: & * P + R.

P5: If P,Q,P',Q' € £>' are such that P = {A U B,Cp,C2,... ,C^},

Q = {A,B,C ,...,C, }, P' = {A U B,C' ...,C!}, 1 K 1 J Q’ = {A,B,C’,...,C!}, then d(P,Q) = d(P',Q').

For each i,j e , set ir. • - {{i,j}}U{{a} | a e J, a / i,j }. 1 i j Notice that ir. . = ir. . for all i,j € ¿8. Set ir = ir. . for 1,1 J,J id i Since ir < P for all P £ £>' > ir is the "trivial" partition.

P6: d(TT,TT. .) = d(TT,TT, ) for all i,j,k,£ € j/, i / j, 1,3 K , X, k / £. 51

LEMMA 2,6. Let d: &' *■ &r m- fl. If d satisfies D2 and P'43 then d satisfies P5.

Proof: Let P,Q,P’,Q' € £>' be as in P5. Set

R = {A,B,/\(A OB)}, S = {A O B,/\(A O B)}. The following relationships are easily verified. Bp(S,P,Q), Bp(S,R,Q), Bp(R,S,P) and Bp(R,Q,P). The remainder of the proof is the same as that of

Theorem 1.2. //

If P,Q € £)' and P and Q are -p-adjacent, then either

P < Q or Q < P. If not, then P A Q / p and P A Q / Q and hence,

B*(P,P A Q,Q). If P and Q are £>' -p-adjacent and P < Q, then it is easy to verify that P = {A,B,C^,C2,...,0^} and

Q = {A U B,C1,... ,Ck), for some A,B£X Given any two partitions,

P and Q, there exists a sequence, P = P^ ,P2, •••>pr+i = Q> dn

£>' such that P^ is & -p-adjacent to P^+1 and Bp(Pi,P^+1,Q) for i = l,2,...,k. Example 2.6 illustrates the construction.

EXAMPLE 2.6. Let / = {1,2 ,... ,6},

P = {{1,2,3},{4,5},{6}}

Q = {{1,4},{3,5},{2,6}}

Then P a Q = it. There are many possible sequences. The following diagram illustrates one: 52

P Q ' ) {{1,2}, {3}, {4,5},{6}} {{1,4},{3,5},{2},{6}}

{{1},{2}, {3},{4,5},{6}} {{1,4},{3},{2},{5},{6}}

7T

Suppose d: x$' +1 satisfies D2 and P'4. Then all distances on /£>' will be completely determined by the values of d on the?0-p-adjacent pairs of partitions. If P and Q are

£)' -p-adj acent and P < Q, then there exists A,B£/ such that

P = {A^Cp...,^} and Q= {AUB,^,...,^}.

Set g = d(P,Q). Since Lemma 2.6 implies d satisfies P5, the value for g Ps independent of the other cells, C^,...,Ck, of P and Q.

Let P € . Then there is a sequence of partitions,

7t = P1,P2,...,Pk = P such that Pp = {ApjijpjjCp. . . ,C*} and

P^+l = {A^ U {j|}, Cp...,C^}, for each i. That is, P^+^ as obtained from P^, by joining a singleton to another cell of P^.

Notice that d(Pp,Pk+p = 6^ {• }• Suppose P,Q £ are such i’ ^1 that P < Q and P and Q are &'-p-adjacent. Let

7t = P P ,. ..,P = P and ir = Q ,. .. ,Q. = Q be sequences as above. 1 4- R 1 J Then since Bn(7r,P,Q), 6D = d(P,Q) = d(Q,ir} - d(P,ir} . Thus, since r A, D d(Q,7r) and d(P,ir) are completely determined by 6^ B's where 53

¡B| =1, I have shown that all -p-adj acent distances, and hence

all distances, are completely determined by the values 6. .. , = 6. ... A,i1j A,1 for A £ i < A, A / 0, A / Z

The following example shows that not just any choice of values for the 6, .'s will result in a distance function on . A,i

EXAMPLE 2.7. Let /= {1,2,... ,n} and suppose d: £)' x x9' -> R satisfies D2 and P'4. Set 6, . = 1 for all admissible A and i. A,i If B = {b^,. .. jb^} £ / and AS/ such that A fi B = 0, then

<5. „ is determined by the 6, .'s. Let A,B 7 A,i R = {A,{bp,.. . ,{br),/\(A UB)}. Then by P'4,

6A B = d(R,{A U B,/\(A U B)}) - d(R,{A,B,/\(A U B)}) = r - (r - 1) = 1.

Thus 6. D = 1 and hence d(P,Q) = 1 for all P,Q £ £)' such that A , D P and Q are £)' -p-adj acent.

Consider the following Hasse diagram of partitions. This is part of the lattice of $'. If P is "below" Q, then P is a refinement of Q. {/= {1,2,3,4}.)

P V Q = {{1,2,3,4}}

P = {{1,2},{3,4}} Q = {{1,3},{2,4}} \ Px = {{1},{2},{3,4}} Q2 = {{1},{3},{2,4}}

P A Q = it

d(P,Q) = d(p,pp + dCP^u) + d(ir,Qp + dCQ1,Q) = 4 but d(P,Q) > d(P,P v Q) + d(P v Q,Q) = 2. Therefore d does not even 54

satisfy D3. This function does satisfy P6, thus the axioms DI, D2,

P'4, P6, and D7 are not sufficient to guarantee that d is even a distance function on . //

Perry's distance function results from setting 6 . = ¡a|. a, i A stronger version of P6, along with the other axioms provides a characterization of this distance function.

P+6: If P,Q € x9' are such that

P = {A,{i},^\(A U {i})} and

Q = {A U {i},if\(A U {i})}, then

d(P,Q) = |A|.

If P+6 is translated into the language of binary relations,

PR and Qr are £)R-adjacent Q0R = {PR | P£ £'}) and |PR A QR| = |a|

Thus P+6 is simply a weak form of Cl. This, along with P'4 and the structure of £>' is sufficient to prove the uniqueness of a distance function on ¿0'.

THEOREM 2.7. Let d: A’ x A’ -> R. If d satisfies P'4 and

P+6, then d(P,Q) = |P^ A for ail P,Q £ A'.

Proof: Let = {PD [ P €0'}. In view of Theorem 2.5, P,Q € P' are P'-p-adjacent if and only if PR and QR are PR-adj acent.

Define dR: 0R x £r + R by dR(PR,QR) = d(P,Q) for all PR,QR£/0R.

Let Pn,Qnex9D be A-adjacent. Then P and Q are A-p-adjacent and d(P,Q) = 6 . Using P'4 and P+6 it is easy to verify that r\ 2 D 6AjB = |A|-|B|. Thus dR(PR,QR) = d(P,Q) = |A|.|B| = |PR A Qr|. 55

Thus dD satisfies Cl. Since d satisfies P'4, dn satisfies

D'4 and by Theorem 1.7, dR(PR,QR) = |PR A QR| for all PR,QR 6 Z>R.

Hence d(P,Q) = |PR A Q | for all P,Q £ £>' . //

The definition of p-betweenness leads us to consider a similar concept in collections of n-trees and other sets of subsets. In

Example 2.1, the information contained in R is, in one sense,

"stronger" than that of P. As a generalization of refinements of partitions, I will introduce the relation of refinement in arbitrary collections, £'. The equivalence of the two definitions for the case £' = Z?' will be verified in Theorem 2.8.

DEFINITION 2.7. Let P £ £' . If {x,y} then the level of

{x3y} in P, L(P,x,y), is the number of elements of P which contain (x,y). That is, L(P,x,y) = ¡{A | (x,y) £ A, A £ P}j.

DEFINITION 2.8. Let P,Q £ £'. P is a refinement of Q, denoted P < Q, if and only if for each x,y £ L(P,x,y) <_ L(Q,x,y)

THEOREM 2.8. Let P,Q € £)’. P is a refinement of Q

(Definition 2.3) if and only if for all x3y €

L(P3x3y) <_ L(Q3x3y).

Proof: Notice that P € -0' and x,y 6 J implies

L(P,x,y) =0 or 1. Suppose P is a refinement of Q (Definition

2.3). If L(P,x,y) = 0, then clearly L(P,x,y) £L(Q,x,y). If

L(P,x,y) = 1, then there exists A £ P such that (x,y) £ A.

Since P < Q, there exists B € Q such that A SB. Hence 56

L(Q,x,y) = 1 and hence L(P,x,y) <_ L(Q,x,y) for all x,y € xf.

Suppose L{P,x,y) <_ L(Q,x,y) for all x,y € /. Let A € P. Then if x € A, there exists B £ P such that x £ B. If A = {x},

A £ B and P < Q. If y £ A, y / x, then L(P,x,y) = 1 = L(Q,x,y).

Thus {x,y} £ C for some C £ Q. Since x € B n C, C=B and hence A £ B. Therefore P < Q in the sense of Definition 2.3. //

DEFINITION 2.9. Let P,Q,R £ £'. If R < P and R < Q, then

R is a common refinement of P and Q. If in addition, for every

S € g' , S < P and S < Q implies S < R, then R is the meet of

P and Q, R = P A Q.

EXAMPLE 2.8. Let / = {1,2,3,4} and If' be the collection of all sets of non-empty subsets of /. Let P = {{1,3,4},{1,2},{2,3}},

Q = {{1,2,3},{1,4},{3,4}}, R = {{1,2,3},{1},{3,4}} and

S = {{1,2,3},{1,4},{3}}. Both R and S are common refinements of P and Q. If P A Q existed, then R < P A Q and S < P A Q.

Now P A Q / R,S since R / S and S / R. The following "level table" verifies these statements and gives the levels that P A Q would necessarily have if it existed. 57

ix,y} P Q R s P A Q 1,1 2 2 2 2 2 2,2 2 1 1 1 1

3,3 2 2 2 2 2 4,4 1 2 1 1 1

1,2 1 1 1 1 1 1,3 1 1 1 1 1

1,4 1 1 0 1 1 2,3 1 1 1 1 1

2,4 0 0 0 0 0 3,4 1 1 1 0 1

Suppose P A Q 6 £' with the levels indicated in the table.

Since L(P A Q,3,4) = 1, L(P A Q,2,3) = 1, and L(P A Q,2,4) = 0, there must exist A,B £ P AQ such that {3,4} £ A and {2,3} £ B, ,

A / B. Since L(P A Q,2,2) = L(P A Q,4,4) = 1, no other elements of P A Q contain 2 or 4. Now L(P A Q,l,2) = L(P A Q,l,4) = 1.

Hence 1 € A and 1 £ B. But then L(P AQ,1,3) ^2, a contradiction.

Thus PAQXC, and meets in S' need not exist.

DEFINITION 2.10. Let P,Q £ S' and x,y 6P and Q agree on {x3y} at level ot if and only if ot = min{L(P,x,y) ,L(Q,x,y)}.

EXAMPLE 2.9. Let {1,2,...,6} and P,Q,R be the 6-trees with Hasse diagrams: 58

By constructing the "level table" for P, Q, and R, it is easy to verify that Q < R and Q < P. Also the level of (x,y) in Q is "between" the level of {x,y} in P and R for every x,y £ Z This relationships suggests a new definition of betweenness for sets of subsets. Theorem 2.9 justifies the use of the same notation as in Definition 2.4.

DEFINITION 2.11. Let P,Q,R £ . Q is p-between P and R if and only if min{L(P,x,y),L(R,x,y)} <_ L(Q,x,y) <_max{L(P,x,y),L(R,x,y)} for all x,y e J. Strictly p-between and £' -p-adjacent are defined similarly and the notation for p-betweenness is identical to that of

Definition 2.4.

THEOREM 2,9. Let P3Q3R€£)’. Bp(P3Q3R) (in the sense of

Definition 2.4) if and only if min{L(P3x3y) 3L(R3x3y)} <_L(Q3x3y) <_ max{L(P3x3y)3L(R3x3y)} for all x3y £ J.

Proof: Suppose Q is p-between P and R (Definition 2.4).

If L(P,x,y) = L(R,x,y) = 1 for x,y £ J, then (x,y) £ A n B for some A £ P and B € R. Since P A R < Q, there exists C £ Q 59

such that A fl B £ C. Hence L(Q,x,y) - 1. Thus

min{L(P,x,y), L(R,x,y)} <_ L(Q,x,y) for all x,y £ Z If

L(P,x,y) = L(R,x,y) = 0, then for every C £ Q, {x,y} C.

Otherwise, since C £A for some A € P U R, either L(P,x,y) = 1

or L(R,x,y) = 1, a contradiction. Therefore L(Q,x,y) = 0 and

L(Q,x,y) £ max{L(P,x,y),L(R,x,y)} for all x,y £ xf. Conversely,

suppose the inequality holds for all x,y £ /. Let A fl B € PAR

for A € P, B £ R, and let x £ A fl B. Then there exists C € Q

such that x£C. If A OB = {x}, then A fl B £ C and hence

P A R < Q. If y £ A 0 B, y / x, then L(P,x,y) = L(R,x,y) = 1

and hence L(Q,x,y) = 1. Thus there exists C' € Q, such that

{x,y} £ C' and therefore C = C' and AHBcC. Hence P A R < Q.

If C £ Q and x € C, then x £ A fl B for some A £ P, B £ R.

If C = {x}, then C £ A and we are done. If y £ C, y / x, then

L(Q,x,y) = 1, thus either L(R,x,y) =1 or L(P,x,y) = 1. Therefore y £ A or y € B and hence C £ A U B. If C <7 A and C B, then there exists y,z £ C such that y / A and z 0. B. Since C £ A U B, y € B and z £ A. Thus L(P,y,z) = L(R,y,z) = 0 which implies

L(Q,y,z) =0, a contradiction since {y,z} £ C € Q. Thus either

C £ A or C £ B and therefore Bp(P,Q,R) in the sense of

Definition 2.4. //

The representation of partitions as binary relations on Z and the equivalence of the betweenness relations in each context, made it especially easy to define a "nice" distance function on . 60

A similar representation for sets of subsets of ¿8 would be desirable

for this same reason.

Let "<" be an arbitrary (but fixed) total order on xf.

For a partition, P, of xf, whenever {a,b} £ A for some A € P,

{a,b} $7 B for any B £ P, B / A. If P £ S' for some arbitrary

collection of sets of non-empty subsets of j/, this need not be

true. A third coordinate is introduced to keep track of how many

elements of P contain {a,b}. If P£ define

P* = {(a,b,i) | a < b, {a,b} £ A for some AGP and 1 £ i £ L(P,a,b)}

(a £ b means a < b or a = b).

Thus (a,b,i) £ P* means (a,b) is contained in at least i

elements of P. If ]xf| = n, then there are 2n subsets of -0).

If P € S', the maximum value for L(P,x,y) is 2n Let

.£* = J x {1,2,...,2n 1). Then P* may be thought of as a binary

relation on jJ* by associating the triple, (a,b,i) with the pair,

((a,i),(b,i)). I will retain the ordered triple notation because

of its simplicity but refer to P* as a binary relation on j/. The

third coordinate is used only to distinguish ordered pairs that would

otherwise look identical. If P € P)' then

P* = {(a,b,l) | {a,b) £ A for some A € P, a £ b). PR is easily

identified with the subset of P*, {(a,b,l) € P* | a / b). Since

(a,a,l) £ P* for each a £ J, in the case S' = £)' , this

information is really not necessary. Thus the two representations,

PD and P*, are essentially the same if P €X0'. R 61

If ’

P* = {(1,2,1), (1,2,2), (1,2,3),(1,3,1),(1,4,1),(1,4,2),(2,4,1), (2,4,2),

(2,3,1), (3,4,1),(1,1,1),(1,1,2),(1,1,3),(1,1,4),(2,2,1),(2,2,2),

(2,2,3),(2,2,4),(3,3,1),(3,3,2),(4,4,1),(4,4,2),(4,4,3)}.

The tree of subsets R = P\{{1},{2}}, which has Hasse diagram:

has representation R* = P*\{(1,1,4),(2,2,4)}. If the strict total order were used to obtain these representations, R* and P* would be indistinguishable. Notice that this representation provides no way of determining whether or not a set of subsets contains 0. For this reason, I have restricted our attention to collections of sets of non-empty subsets of J. //

THEOREM 2.10. Let P3Q3R € £’. Bp(P3Q3R) if and only if

B(P*3Q*3R*). 62

Proof: Suppose Bp(P,Q,R) and let (a,bji) £ P* 6 R*. Then

i £ min{L(P,a,b),L(R,a,b)} and therefore i £L(Q,a,b). Thus

(a,b,i) € Q*. If (a,b,i)£ Q*, then

i £ L(Q,a,b) £ max{L(P,a,b), L(R, a,b) } and therefore (a,b,i) £ P* u R*

Thus B(P*,Q*,R*). Conversely, suppose B(P*,Q*,R*) and let

a = min{L(P,a,b),L(R,a,b)} for some a,b £ S. Then

(a,b,a) £ P* H R* which implies (a,b,a) £ Q*. Thus a £ L(Q,a,b).

If y = L(Q,a,b), then (a,b,y) £ Q* £ P* U R*. Hence

Y £ max{L(P,a,b),L(R,a,b)}. Therefore Bp(P,Q,R). //

Let d: S' x £' R. The following axioms are extensions of

P4, P5, and P+6.

S4: If P,Q,R £ S' then Bp(P,Q,R) if and only if

d(P,R) = d(P,Q) + d(Q,R).

S'4: Similarly.

P+5: If P,Q,P' ,Q' e S ' are such that P = {A U B,C]L,C ,.. . »cp,

Q = {A,B,C1,...,Ck}, P’ = {A U B,q,...,Cp,

Q' = {A,B,C|,...,Cp, then d(P,Q) = d(P',Q').

S6: If P,Q £ S' are S' -p-adjacent, then d(P,Q) = |P* A Q*|.

If P = Q, then d(P,Q) = 0. The proofs that D2 and S'4 imply S5 and P+5 are similar to

Lemmas 2.5 and 2.6. The following theorem gives a characterization of a new distance function on Jf'. It will be denoted by d* to distinguish it from the symmetric difference distance function on £' . 63

THEOREM 2.11. Let d*: g' x if d* satisfies S’4 and S6 then d*(P3Q) = \P* A for all P3Q € S’. Furthermore d* satisfies D1-DZ3 S4, S53 and P+5.

Proof: Let g* = {P* | Peg’} and let d: £* x £* -> R be defined by, d(P*,Q*) = d*(P,Q) for all P*,Q* € g*. In view of Theorem 2.10, d satisfies D'4 and Cl, thus d(P*,Q*) = |P* A Q*| for all P*,Q* € g* (Theorem 1.7). Hence d*(P,Q) = |P* A Q*| for all P,Q £ £'. Since d also satisfies D1-D4, d* satisfies

D1-D3 and S4. Furthermore, d satisfies T5 from which it is easy to deduce that d* satisfies both S5 and P+5. //

EXAMPLE 2.11. Let /,P,Q,R be as in Example 2.9. The following tables give the distances between P, Q, and R for the distance functions of Theorems 2.2 and 2.11.

d I P Q R d* 1 P Q R P045 P 08 14 Q 0 1 Q 0 6 R 0 R 0

Notice that Bp(P,Q,R) and B(P,Q,R) both hold in this case. Next let /, P, Q, R, and S be as in Example 2.1. Then the distances are

d 1 P Q R S d* 1 P Q R S P0121 P 0336 Q 0 12 Q 0 5 9 R 0 2 R 0 3 S OS 0

In this case B (P,R,S) while d(R,S) = 2, d(P,R) = 2 and d(P,S) = 1. Also B(P,Q,R), but d*(P,Q) = 3, d*(Q,R) = 5 and d*(P,R) = 3. // 64

The betweenness concepts upon which these two distance functions are founded are quite distinct. The resulting distances reflect this distinctness. Therefore the choice of a distance function should be made according to the type of betweenness that best agrees with your intuition in the application at hand. CHAPTER 3

A CONSENSUS FOR DISCRETE STRUCTURES

The problem of defining a "consensus" for a collection of

discrete structures arises in many contexts [1,2,5,16,18,19,21,26,

31]. In the conclusion of an article dealing with the assessment

of cladograms, B. Baum raises the question, "Given an array of

plausible cladograms from various methods, how does one make a

plausible speculation about evolutionary history from them?" [3].

E. Adams, in considering tree posets and n-trees defines the

consensus of two or more such structures as "a tree representing

only that information that is shared by all of the trees" [1]. A more liberal definition of consensus might be a tree representing

the information shared by a majority of the trees. Kemeny and

Snell's approach to a consensus for collections of rankings (types

of preference relations) involves distance concepts [18]. Their results support the more liberal definition of consensus. K. Bogart

extended Kemeny and Snell's ideas to collections of preference

relations [5], antisymmetric orderings [6], and in a joint paper with Weeks, signed digraphs [33]. I will adopt the same approach in my consideration of consensus for collections of discrete structures

Consider a two-outcome voting game in which each person votes for either A or B. An individual vote can be represented by

-65- 66

one of the preference relations in £ = {{(A,B)},{(B,A)}}. These

are the only possible votes. The collection of votes cast by the

group is called a "profile". A possible profile for a group of

three voters is [{ (A,B) },{ (A,B) },{ (B,A) }] . Square brackets, rather

than braces, are used to denote a profile to allow for repetition

of votes. A decision rule is a function which assigns a unique

element of C to each profile. This is called the "collective

decision". D. Rae defines a "best" decision rule as one which

maximizes the probability that the collective decision will agree

with any individual's vote [24]. One possibility for a best decision

rule for an odd number of voters, is one that decides the collective

decision by majority rule. Since this is the decision which agrees

with most of the votes, this decision rule also chooses a "consensus"

for each profile.

In the two-outcome voting game, each vote either agrees or

disagrees with the collective decision. The concept of "agreement"

is less well-define in multiple choice situations, for instance where the voters are asked to rank three or more alternatives in

order of preference. In these situations, distance functions are important tools for measuring the amount of agreement.

The following definitions will apply to collections of binary relations, £, and also to collections of sets of subsets, £', although they are stated only in the context of binary relations.

DEFINITION 3.1. A profile in £ is a collection of elements of G with repetition allowed. If P is a profile in G and

Pj,P ,...,Pk are the elements of P, then P = [P^,P2,...»P^]. 67

| P| denotes the number of elements in P, P <= £ means

{Pj,.. . ,Pp £ g, and P^ £ P means P^ € {P^,. . . ,P^}.

DEFINITION 3.2. Let P = [P ,P2,...,Pk] be a profile in £*.

M € £ is a median for P in Z (with respect to a distance function, k k d, on £) if and only if £ d(M,P.) <_ £ d(M’,P.) for every i=l 1 i=l 1 M' € £.

The distance function under consideration will be clear from the context.

A median of a profile is an element of £ that "maximizes the amount of agreement" between it and the elements of the profile in the sense of minimizing the total distance between itself and the elements of the profile. In the two-outcome voting game, a decision rule which assigns to each profile, a median for the profile, is a

"best" decision rule in the sense of Rae.

Since the sets under consideration are all finite, and the distance functions take only finite values, a median must exist for any profile in £.

DEFINITION 3.3. Let P be a profile in £. The majority rule of P, denoted 7>f(P) , is defined by

x € 7^(P) if and only if x is an element of a majority

of the elements of P.

DEFINITION 3.4. Let Reg and Q £ / x/. r is minimally close to Q in £ if and only if for each S <= / x xf, B*(Q,S,R) implies S / g. 68

K. Bogart considered several questions about medians in the context of preference relations and antisymmetric orderings. The next theorem is suggested by his results in [6]. Medians are with respect to the symmetric difference distance function on G.

THEOREM 3.1. Let P be a profile in G and M be a median for P in G. If Q £ G and B(tv((P),Q,M), then Q is a median for P in G. So if 7>f(P) £ G then %((P) is a median for P in G. If ^(P) £ G and there does not exist a,b £ ■& sueh that P. £ P}| = -Jy-L then ^f(P) is the unique | {i | (a,b) £ P. and 'b "Is ¿j median for P in G. If it is not the unique median, then ff(P) £ M' for any other median, M', for P in if.

Proof: For each R £ G and a,b £ define the indicator function

1; if (a,b) €. R h.b™ ' 0; otherwise

I Fa,b'R’ - h(S)i Notice that d(R,S) a,b (a,b)e/xj Let P = [P1,P2,...,Pk] be a profile in £, and let M,Q £ G such that B(V>£ (P) ,Q,M) . Fix a3b£.

x = x(a,b) = - la.bfi’l

y = y(a,b) = JJ’a.b™ ’ ’a.b^i’ 1=1

z = z(a.b) - niiibW - Ia;btpp|. 69

If Ia,b('^('P)-) = Ia,b(-M-1’ then since BOf(pbQ>M),

Ta,b^ = Ta,bW = Ia,b(^(Pn

in which case x = y = z. If I , / I ,(M) then since a, d a, d I k(Q) = 0 or 1, either x = y or y = z. a 9 d Suppose I , (yrt (P)) = 1 and I ,(M) = 0. Since (a,b) £ X(p) a j d a, d

■{i I ^b^P = bPi e p}l = |{i 1 (a’b) € pi}| > I hence x <_ < z. Also, since x = y or y=z, x <_ y <_ z and one of the inequalities is strict.

Suppose I , (7>f(P)) = 0 and I , (M) = 1. Then since a, d a, D (a,b) 0 W),

l{i I ^b^P = ljPi € P}l = '{i • (a’b) € Pi}| - I

k Therefore x <_ £ z with equality holding if and only if I{i | (a,b) € P.}| = | .

Again since x = y or y=z, x <_ y <_ z. Also notice that if (a,b) £7^(P) a M and x = z, then (a,b) £ M\fl((P) .

Summing x, y, and z over all a,b £ J, we obtain the inequality: k k k I d(^(P),Pi) < Z d(Q,P ) < I d(M,P.) i=l i=l i=l k k If M is a median, then Z d(M,P.) = Z d(Q,P.) and therefore i=l 1 i=l 1

Q is also a median for P in S. If 7^(P) 6 £, then

B(7^(P) (P) ,M) implies ?^(P) is a median. If it is not the unique median and M’ is any other median, then for each 70

(a,b) G M' we have (a,b) £ M’\7>{(P). Thus 7Jf(P) £ M’.

Furthermore (a,b) is in exactly half of the elements of P. //

COROLLARY 3.1. Let P be a profile in If such that |p| is

odd. If 7>f(P) € {f3 then 'fif(P) is the unique median for P in £.

Proof: Since JP} = k is odd, there does not exist a,b € Z,

such that |{i | (a,b) 6 P^,P^ p}| ~ y • Therefore by Theorem 3.1,

**f(P) is the unique median for P in £. //

COROLLARY 3.2. Let P be a profile in If. If 77(fP) 0 £ then

P has a median M £ if such that M is minimally close to "TJffP)

in C.

Proof: Let M^ be a median for P in ¡f. If is not

minimally close, choose £ £, such that B* (7?f (P) ,M2 ,Mp . Then

by Theorem 3.1, M2 is also a median for P in £. If NL £ £

(i 2) has been chosen, choose M^+^ £ £ such that B* (7^(P) ,FL + ,FL).

FL+i is also a median. After finitely many steps (at most

(P) A M | - 1) no such element of £ can be chosen. That is

{M 6 g | B*(7% (P),M,M.)} = 0 for some j 1. Thus is

minimally close to 7?f(P) in £ and NL is a median for P in £. //

EXAMPLE 3.1. Let Z = {1,2,3} and P [P ,P2,P3] be a profile

in T9 where P]/P2’P3 ^ave Hasse diagrams:

Pf ’1 P2: P3:

2 1 3 71

7^(P) = {(3,1),(1,2)} / 7°. It is however, by Theorem 3.1, the unique

median for P in the collection of all binary relations on Z.

According to Corollary 3.2, there is a median MG/5 that is

minimally close to 7*f(P) in Z. Let M = {(3,1)}, then 3 3 2 d(M,P.) = 7. Since 2 d(7f(P),P-) = 6, and no other binary i=l 1 i=l 1 relation can attain this sum, M must be a median for P in ~P.

{(1,2)} is also a median and therefore the median for P in / is

not unique. //

Similar results hold for collections of sets of subsets of z.

First we will consider medians with respect to the distance function

d on The proof of the following theorem and its corollaries

are simply translations of the proofs of Theorem 3.1 and Corollaries

3.1 and 3.2.

THEOREM 3.2. Let P be a profile in G’ and M be a median

for P in Gr. If Q € G' and B(Hf(P),Q,M), then Q is also a

median for P in G'. If I^(P) £ G’, then "^(P) is a median for

P in G’• If I>((P) £ £' and there does not exist A £ jS such that |{i | A £ P.,P. £ P}| = -LyL , then ^(P) is the unique

median. If it is not unique and M' is any other median, then n((P) £ M'.

COROLLARY 3.3. If P is a profile in G' and |P| is odd

and TffP) £ Gr, then 7%(P) is the unique median for P in G'- 72

COROLLARY 3.4. If P is a profile in S' and 7f(P) 0 S’,

then P has a median M & S’ sueh that M is minimally close to

70(P) in S'. *

For collections of preference relations, tree posets, and

n-trees (represented as binary relations), the majority rule of a

profile is not necessarily an element of f3, If, or Sf. The

subset representation of n-trees behaves rather nicely as the

following theorem shows.

THEOREM 3.3. If P is a profile in Jf', then T0(P) £ IP.

Proof: Let P = [Pn,. . . ,P, ] S . Then 0 0 P. , / £ P. and

(a) £ P^ for each a £ ^, i=l,2,...,k. Thus 0 0 7^(P),

J £7^(P) and {a)£7^(P) for each a £ xf. If A,B€7?(P) and

A D B / 0, then there is at least one i such that P^ € P and A

and B £ P.. Since P. £ fP and A,B £ P. and A O B / 0, either 11 1 A c b or BSA. Therefore 7>/(P) e#'. //

COROLLARY 3.5. If P is a profile in £', then 7f((P) 6 £’.

Proof: The proof of Theorem 3.3 contains all the steps needed

to show 7>^(P) £ £'. II

COROLLARY 3.6. If P is a profile in Ip($') then tJ{(P) is

a median for P in IP(.£').

Proof: By Theorem 3.3 (Corollary 3.5) M(p) Glf’CS') and hence

by Theorem 3.2, Z^(P) is a median. // 73

If P is a profile in 7>f(P) need not be an element of Jf.

The corresponding profile P' in Jf' however, has the property

that TTf(P') £jf'. It would be nice if the ordered pairs representation

of 7?f(P') were a median for f in xK. The following example shows

this need not be true.

EXAMPLE 3.2. Let Z= {1,2,...,5} and P= [P1,...,P?] be

the profile in Jf such that P^,...,P have the following Hasse {1} {2} {3} {4}

P P P • 1’ 2’ 3‘

Let P' = [Pj,?2,•••»P7] de the corresponding profile in Jf'. By

Theorem 3.3, #f(P’) € Jf' . In fact, 7>?(P') = P^. Since |P’| is

odd, P^ is the unique median for P' in Jf' . Notice that

•>^(P) = P4\{ ({1,2,3,4},{1,2})} 0Jf. In the collection of all binary

relations on the power set of xf, ^(P) is the unique median and 7 £ d(7^(P),P.) = 31. Both P4 and P$ are minimally close to i=l 7 7 7^(P) in Jf. Since £ dfP^Pp = 36 and £ d(P5’PP = 35, P4 i=l i=l is not a median for P in Jf. //

Finally we will consider medians for profiles in £' with

respect to the partition metric, d* on £'. If P is a profile 74

in G’, 7^(P) is not necessarily a median for P with respect to

d*. It need not even be minimally close. The relation of p-between-

ness requires a different concept of "majority rule" in order to

attain results similar to Theorems 3.1 and 3.2 The following

example illustrates the problems inherent with 7^(P) when P ££’.

EXAMPLE 3.3. Let Z = {1,2,...,8}.

Px = {{1,2,3},{4,5} ,{6,7,8}}

P2 = {{1,3},{2,4},{5,6,7},{8}}

P3 = {{1,4},{2,3,5},{6,7},{8}}

P = [P^,P2,P3] is a profile in x9’. In the usual sense of majority

rule, 7?f(P) = {{8}} which does not capture the idea of a consensus

in this instance. Notice that all three partitions agree on pairing

together 6 and 7. If we consider the binary relations P*, P*,

and P* and set P* = [P*,P*,P*], then

7<(P*) = {(1,3),(2,3),(6,7)} U {(i,i) | i € Z} (the third coordinate

is always one and is therefore suppressed). If M were a set of

subsets of Z such that M* ='/f(P*), then {1,3} and {2,3} € M.

But then 3 occurs twice which is impossible since (3,3,2) 7?f(P*).

Thus 7^(P*) is not the representation of any set of subsets of Z.

Nevertheless, X(p*) is a median for P* in the collection of all binary relations on Z (with respect to d). Furthermore, by

Corollary 3.2, there is a median for P* in £* which is minimally

close to 7>f(P*) in $*. If M = {{1,3},{6,7},{2},{4},{5},{8}} then 3 M* is minimally close to ^(P*) in x9* and \ d(M*,Pt) = 13. i=l 75

3 Since £ d(7?((P*) ,P*) - 12, and this is unique, M* is a median i=l 1 for P*, and hence M is a median for P in £>' (with respect

to d*j. In this case, M is not unique since

{{2,3},{6,7},{1},{4},{5},{8}} is also a median.

DEFINITION 3.5. Let P be a profile in g'. If there exists

a set of subsets M such that M* - 77f(P*), then M is the

p-majority rule for P. If Meg', then it is the p-majority

rule for P in Jf’.

EXAMPLE 3.4. Let Z = (1,2,3,4} and P,Q,R £ Jf' with Hasse

diagrams:

Let S = [P,Q,R]. Then 7<(S) = R. If S* = [P*,Q*,R*], then

■^f(S*) = P*. Thus P is the p-majority rule for S in Jf', while

R is the majority rule for S in Jf' . Computing distances,

d(P,Q) + d(P,R) =3+1=4 d*(P,Q) + d*(P,R) = 4+3 = 7

d(R,P) + d(R,Q) =1+2=3 d*(R,P) + d*(R,Q) = 3 + 7 = 10.

THEOREM 3.4. Let P be a profile in Jf’ and N £ if ’ be a

median for P. If Q € Jf’ such that BC?>£(P*)3Q*3N*), then Q

is a median for P in g'. (Medians are with respect to d*.) 76

Proof: Let P = [P^,...,Pk] Q S' and N £ S' be a median for

P. Suppose Q £ S' such that B(7^(P*),Q*,N*). Set

= {R* | R ££'} and P* = [P*,...,P*]. Then k k k k Z d(N*,P*) = Z d*(N,P.) < Z d*(R,P.) = Z d(R*,P?) for any i=l 1 i=l 1 i=l 1 i=l 1 R* € (£')*. Thus N* is a median for P* and by Theorem 3.1, Q* is also. Since d*(Q,Pp = d(Q*,P?), Q is a median for P in £'. //

COROLLARY 3.6. Let P be a profile in S' and suppose M is the p-maj ority rule for P. If M £ S ’, then M is a median for

P in S’- If there does not exist a,b £ J such that

] {i | {a,bj £ A,A £ £ P} | = , then M is the unique median for P in S'. If M 0 S' then there exists a median for

P in S' among the elements of S'3 minimally close to M in S’.

Proof: Let P = [P^,...,Pk] £ S' and M be a set of subsets such that M* = 7>^(P*). Let N be a median for P in S'. If

M £ S', then B(M*,M*,N*) and by Theorem 3.4, M is a median for

P in S'. If there does not exist a,b € xf such that

{i | {a,b} £ A,A £ Pp’Pp £ p}| = > then there does not exist a,b £ xf such that |{i | (a,b,j) £ P?,P£ € P*}| = j and hence by

Theorem 3.1 ?y(P*) = M* is the unique median in (£’)*, so M is the unique median for P in S'• If M 0 S', then Corollary 3.2 implies there exists Q* G (£')* such that Q* is a median for P* and Q* is minimally close to M*. Thus Q is a median for P in

S' and Q is minimally close to M in (with respect to p-betweenness). // r 77

COROLLARY 3.7. Let P be a profile in S'. If the p-majority

rule for P does not exist, then there exists a median, Q, for

P in G’ sueh that Q* is minimally close to "?>((P*) in (£')*.

Proof: Since fl((P*) (£’)*, Corollary 3.2 implies there

exists a median, Q*, for P* in such that Q* is

minimally close to 7>£(P*) in (£’)*. But then Q is a median

for P in 5'. //

EXAMPLE 3.5. Let Z,Pj,---,Py be as in Example 3.2. Then

■#f(P*) = P* so P[_ is the p-majority rule for P in Jf’. Since

fpj = 7, this is the unique median. Notice that 7 7 1 d*(P5,P.) = 24 and \ d*(P4,P.) = 39. // i=l i=l

If the problem of finding a consensus for a collection of

discrete structures is viewed as equivalent to that of finding a

median with respect to an appropriate metric, then the structure

obtained by majority (p-majority) rule is a key step in defining a

consensus. In case the majority rule is one of the discrete struc­

tures under consideration, it is a median for the collection, and in

many cases it will be unique. If the majority rule does not yield a

member of the collection, the problem of finding a median is still

difficult. Bogart suggested using the betweenness relationships

to restrict the number of structures one needs to consider in

searching for a median. In this connection the concept of "minimally

close" seems to be important. Determining the collection of minimally

close elements in the various contexts still remains to be done. BIBLIOGRAPHY

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