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Neutral consensus functions
Mathematical
Social
Sciences
231
25 (1993) 231-250
North-Holland
Neutral
consensus
functions
Gary D. Crown Wichita State University, Wichila, KS 67208, USA
Melvin F. Janowitz Universiry of Massachusetts, Amherst, MA 01003. USA
Robert C. Powers University of Louisville, Louisville, KY 40292, USA Communicated
by F.W. Roush
Received August
1992
Neutral consensus methods have been studied in detail by a number of authors in specific contexts. This paper introduces a general mathematical model that allows for a unified approach to their study via the notion of stability families. Special attention Key words: Consensus
method;
is paid to conditions
neutral;
that produce
abstract
versions of Arrow’s Theorem.
stability family.
1. Introduction The need to summarize a collection of structures into a single object has arisen in such diverse areas as social choice theory, evolutionary biology, numerical taxonomy, economics, psychology and classical statistics. Such a procedure is often called a consensus technique or a consensus method. It has long been recognized that because consensus theory has independently been developed in many diverse disciplines, there is a need for a general mathematical model that will produce results having wide-ranging interpretations. One such approach is by means of ordered sets. The present work has its genesis in two recent approaches to these issues by means of various ordinal structures (see Barthelemy and Janowitz, 1991, as well as Monjardet, 1990). For further background the reader might consult the references mentioned in these sources. It is worthy of mention though that much Correspondence to: M.F. Janowitz,
Department
of Mathematics,
University
of Massachusetts,
MA 01003, USA.
01654896/93/$06.00
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1993-Elsevier
Science
Publishers
B.V. All rights
reserved
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232
G.D.
Crown
et al. / Neutral
consensus
functions
of the activity in consensus theory stems from Arrow’s seminal work (Arrow, 1962). Essentially Arrow produced a number of seemingly reasonable axioms that a consensus method on weak orders should satisfy, and then proved his celebrated ‘impossibility’ theorem - that no such consensus method could exist. Since then much research has been devoted to extending Arrow’s Theorem to other contexts, and seeking ways to modify the axioms underlying Arrow’s Theorem so as to avoid its rather distressing consequences. Our general approach will be to use an idea of Barthelemy and Monjardet (1981), and consider the construction of a consensus object as a process where certain ‘building blocks’ or ‘bricks’ are glued together in a suitable manner. Although we shall keep the basic setting arbitrary, we will when appropriate show how our results generalize some of the standard theory on consensus. A major goal will be to determine just what it is about the ‘building blocks’ that produces
analogs
of Arrow’s
Theorem.
2. Notation The basic notation and terminology of the paper will be established here. Before one can even think of performing a consensus, one must identify the set of objects to be summarized. We agree to let X denote this set. Although some of the relevant examples are finite, we impose no such restriction on X; indeed, we agree in general that X shall denote a possibly infinite set having at least three members. In social choice theory, X might represent subsets of the set of alternatives under consideration; in voting theory, X might denote the linear orders or the weak orders on the set of candidates, etc. V will denote a fixed (possibly infinite) indexing set having at least two members. When V is finite, it will be useful to let V= { 1,2, . . . , n} for some positive integer n > 1. Think of V as the set of voters or the set of judges or the set of committee members who are responsible for producing the consensus object. P= XV will denote the set of mappings from V into X. A member of P is called a profile. In voting theory, one can think of an element of P as the collection of ballots in an ‘election. For V finite, a typical profile will be 7~= (x1,x2, . . . ,x,) with xi E X for all i. In the infinite case, one can think of a profile as an indexed family where rc((r)=x, for each a~ Y. 7r =(x,),, y or else as a function 7c: V/-X, A consensus function or a consensus method may now be taken as a mapping F: P - X. Other authors who have considered consensus methods have allowed the set V to vary, and have examined the effects of voters being added or deleted from I’. They have also considered consensus functions that produce non-empty subsets of X rather than single elements, or even take values in some set other than X. Issues such as these will be considered in a later paper. Finally (as mentioned earlier), we introduce a set S of building blocks or bricks that can be used to construct the output consensus object (see Barthelemy and Monjardet, 1981). The general approach will be to study consensus methods in terms of
G.D. Crown et al. / Neutral consensus funcrions
233
how they interact with these building blocks. Thus a stability family on Xis defined to be a pair (S, y), where S is a given set of building blocks and y :X - P(S) is a mapping. Here are some examples that the reader should keep in mind. 1: Binary relations. Let Y be an underlying set, and X some set of binary relations on Y. Let S = Y x Y, and for each R EX, let y(R) = {(x, y) : xRy} . So the idea here is to form a consensus relation F(rc) from a profile 7~by determining the pairs (x, y) that would belong to the consensus. Of course the problem is compounded by the need to have F(n) E X. Alternatively, one could let y(R) = {(x, y) : xRy fails}. Example
Example 2: Finite lattices and semilattices. Let X be a finite lattice. Then every element of X is a join of sup-irreducibles and a meet of inf-irreducibles. Thus we can take S = either the set of sup-irreducibles or the set of inf-irreducibles of X, and in the first instance define
y,(x)=(bES:blx}; in the second instance, take yz(x)={bES:bzx}. We could also use the dual of either of these examples, or even take S = (s E X: s is sup-irreducible or inf-irreducible), with v(x) = y,(x) U ya(x). Note here that we do not regard the smallest element of a lattice to be sup-irreducible, since 0= V {xEX:X
of Y satisfying: l
YET.
l
0 is not in T.
l l
{y}ETfor
aIlyEY.
A,BE T implies A nBE {O,A,B}.
The members of T are called clusters, and the members other than X and the singletons are the proper clusters. We can now take S = B(Y), and for each n-tree T, let y(T) denote its proper clusters. Thus a consensus method F would construct F(n) by choosing its proper clusters. Similar examples may be formed by consider-
G.D. Crown el al. / Neutral consensus fundons
234
ing other types of families (Buneman, 1971).
3. Consensus
families
of subsets of a set. Among
and neutral consensus
them are thephy/ogenetic
trees
functions
Let F be a consensus function, (S, y) a fixed stability with the notation introduced in Section 2.
family
on X and continue
Definition 1. (i) For each XEX, let 7cXdenote a profile whose entries are all X. (ii) F is called neutral if for all profiles rc and rc’ and for all s, S’E S, it is true that if {BE v:sE)+r(a))}={crE
V:s’E)@‘(a))$,
then SE y(F(n))
if and only if S’E y(F(n’)).
(iii) F is called monotone-neutral it is true that if
if for all profiles
71 and 71’ and for all S,S’E S,
then s E y(F(~c)) implies
s’ E y(F(n’)).
Lemma 2 contains a useful and well-known fact about neutral and monotoneneutral consensus functions that will provide a central theme for the remainder of the discussion. 1991, Lemma 3 and 4). The consensus 2. (Barthtlemy and Janowitz, method F is neutral of and only if there is a collection C@)F of subsets of V having the property that
Lemma
s~y(F(z))
if and only if {a:sEy(x(a))}
It is monotone-neutral if it is neutral and the family sense that A E aF, A C_B implies BE BF.
E !SF. kDF is an order filter in the
Order filters are called federations in Monjardet (1990). The idea now is to relate properties of y with the nature of the possible collections gF of subsets of V that are associated with neutral consensus functions. The start of this work is in Monjardet (1990), and it played a role in Barthelemy and Janowitz (1991) in the context of Example 2. We would like here to view the material in as genera1 a setting as possible, and (when feasible) in each instance to obtain necessary and sufficient conditions for a given property to hold.
Remark
3. It seems reasonable
to require
that
S= UxeX v(x). This
follows
from
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235
the fact that if se y(x) for any XE X, then s is not going to have any effect on a consensus function F. If y is not one-to-one, then it can only determine a neutral consensus function up to the equivalence relation induced by y. In this case it seems appropriate to just idenfity x withy in the case y(x) = y(y), and work with the quotient set obtained by this identification. Once this is done, y induces a one-to-one mapping on this set. In view of this, we shall henceforth simply assume that for any stability famiiy (S, y), the mapping y is one-to-one. Remark 4. (i) The notation
will be greatly simplified, and many arguments will become easier to understand if we simply identify each XE X with the corresponding subset y(x) of S. Accordingly, we herewith change our setting: let S be a set having at least three members, and XC 9(S), leaving the terminology involving V unchanged. (ii) For a E S and II E P, it will be convenient to let a, = {a E V: a E n(a)}. (iii) Let $2 be a family of non-empty subsets of Vsuch that VE 97. For each rc E P, we let 9, = {a E S : a, E 95}. Finally, we may define a relation 0 from S to P by the rule aOx if and only if a, E 9, and note that 0 induces a galois connection between B(S) and 9(P) by the rules: a(T)={7rEP:a&r
for all aET},
jl(Q)={aeS:aOn
for all ZEQ},
for all subsets Tc S and Q c P. We assume that the reader has a basic knowledge of the theory of galois connections. The results we shall use can be found in any standard text on lattices and ordered sets. In particular, the subsets Tc S and Q c P will be called closed in the case
T = PM”)), Lemma Proof.
Q = d.NQ)).
5. For each x E X, x is closed.
We simply note that
B(dx)) =PW%J
= B(dD({n,)))) =P({7c,)) =x.
Definition 6. The set 91 of subsets of V is
~27as a neutral
n,n(a) c gd, c n, n(a).
0
G.D. Crown el al. / Neutral consensus functions
236
Before proceeding, we repeat the connection between the two approaches to neutral consensus functions. On the one hand, we have a set X and a stability family (S, y). By identifying each element of X with its image under y, we produce a family of subsets of S that induces a neutral consensus method in the sense of Definition 6. Suppose, on the other hand, that $27is a consensus family in the sense of Definition 6. Then for each rc E P, ~2)~E X. We may define a mapping F: P - X by the rule F(x) = 9),, take v(x) =x for all XE X, and note that (S, y) is a stability family for F and that F becomes a neutral consensus function in the usual sense, in that s E F(x) if and only if s, E %,. Remark 7. We agree to order X by set inclusion on S. If X is closed under unions,
then it clearly forms a join-semilattice; similarly, closure under intersection makes it into a meet-semilattice. If it is closed under both unions and intersections, then it is a distributive lattice. It is important to notice that X can be a distributive lattice without it being closed under both unions and intersections. To illustrate this concretely, we ask the reader to recall that a weak order on a set Y is a binary relation W that is reflexive, transitive and complete in the sense that for all x, y E Wit is true that x Wy or y Wx. Consider the set X of weak orders on a set Y having at least four elements, taking as a stability family Y x Y with y(W) = {(x, y) : x Wy} for each weak order W. When ordered by inclusion, X forms a join-semilattice that is not a lattice, but which has the property that each of its principal filters is a distributive lattice. Since the union of a pair of weak orders need not be transitive, it is not true that y(W,VW2)=y(Wl)Uy(W2) for all W,, W,EX. Definition 8. There is an interesting notion of duality related to all of this. Let X be a family of subsets of S with V defined in the usual manner. Let
X*={X’:XE~),
where x’=S\x,
p*=(X*)&{n*:
n E P} , where n *(cr) = [n(o)]‘.
If ‘3 is a consensus family on X, define the dual family 9*=(/I
ka* by
c V:A’$kD}.
It will also be convenient to define the complementary family of 8 by the formula a’=&4
c I’:A$G!r}.
Note that the mapping A - A’ is a dual automorphism of 9(V) that maps 91’ onto a*, and G@onto (g*)‘. The next theorem gathers up the salient facts about this notion of duality. 9. Let 43 be a consensus family on X. Then: (i) 52J*is a consensus family on X*. (ii) For each 7c* E P*, a,*, = (~3~)‘. (iii) 5%= a**.
Theorem
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Crown
et al. / Neuiral
consensus functions
231
(iv) 9 is monotone-neutral if and only if $B* is monotone-neutral. (v) $9 is an ultrafilter if and only if LB* is an ultrafilter. In fact, if this is true, then G3= L@*. (iii) will each follow directly from (ii), so we concentrate on (ii). We simply note here that for each a ES,
Proof.
our efforts
This shows that aE $B,*,*ae8,, thus establishing (ii). (iv) This is an immediate consequence of the fact that GZJis an order filter if and only if $2~‘isan order ideal, and this in turn is equivalent to GB*being an order filter. (v) To say that 9 is an ultrafilter is equivalent to saying that &&is a maximal lattice ideal, and this in turn is equivalent to saying that 8* is an ultrafilter. To complete the proof, note that if 9 is an ultrafilter, then AEGB iffA’$%7 iffAED*.
0
Remark 10. (i) When X is either closed under unions or intersections,
a number of concrete examples may be constructed. Here is the underlying principle. Let $3, and $_Qbe consensus families on X. Let g3 = 6BD1 U C@*,and g4 = 9, fl D,. If X is closed under union, then G@J~ is a consensus family and for any profile 71, 5B3,= * similarly, if X is closed under intersection, then &B4is a consensus g1, u a,, family with G2J4, = G3’r.fl $B2,. Since X is closed under union if and only if X* is closed under intersection, these two constructions are dual to each other under the construction outlined in Theorem 9. (ii) Suppose X is closed under unions, and let A be a finite nonempty subset of V. Then 22l={C~ V:aEC for some crEA} is a consensus family, and for each profile z, we have
can= u
7c(a).
asA
(iii) Suppose X is closed under intersections, a=(Cr
with A as in (ii). This time, let
V:AcC}.
and note that a,=
n
77(a).
CIEA
(iv) Finally, assume X is closed under both unions and intersections. Let V be finite, and assume $2 is an order filter of V. Then $8 is a consensus family on X.
G.D. Crown et al. / Neutral consensus functions
238
If A,,A*,
. ..) Ak are the minimal
elements
of $3, then
for any profile
rc,
4. Complements We continue with the notation introduced in Remark 4, and consider the question of complementary subsets of I/ either being or not being members of a consensus family G@. Lemma 11. Let %! be a consensus family on X. (i) If there exists a subset A of V such that both A and A’ are in $3, then X is
closed under unions. (ii) If there exists a subset A of V such that neither A nor A’ are in 525,then X is closed under intersections. Proof.
(i) Let x, y E X, and define
z(a)=x
for aeA
n(a)= y
for aEA’.
a profile
rc by requiring
and
IfaEx\y,thena,=AE4);similarly,aEy\ximpliesa,=A’E~; forces a,=VECB. Since ~J~CXUY, it follows (ii) This is dual to (i). Cl
finally,aExny that
g,=xUy~X.
The next two theorems further develop this natural connection between complementary subsets of V and whether or not X is closed under unions or intersections. Theorem
A’$8
12. Every consensus family 9 on X has the property that A E 9 implies if and only if X is not closed under unions.
Proof. Assume the existence of x, y E X such that XU y 13X. By the contrapositive to Lemma 1 l(i), A E $2~implies A’b 9~. Suppose on the other hand that X is closed under union. Let A c V be non-empty. Choose fixed elements a E A, a’E A’, and let $2 be the collection of subsets of V that contain a or a’. Notice that both A and A’ are members of 93, so we need only show that 23 is a consensus family. But this 0 is easy to do. For any profile rr, we simply note that gD, = n(a) U n(a’) E X. Corollary 13. There exists a consensus family C?ZJ on X that contains complementary subsets of V if and only if X is closed under unions.
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239
The next results follow by applying Theorem 12 to X*. Hence they are dual to Theorem 12 and Corollary 13. Theorem
12*. Every consensus family 91 has the property that for every subset B implies B' E 9 if and only if X is not closed under intersection.
of V, B$9 Corollary
13*. There exists a consensus family 9 on X whose complement contains a complementary pair of subsets of V if and only if X is closed under intersections. Corollary 14. Assume X is not closed under intersection or union. Then for any con-
sensus family Q7on X, and for any subset A of V, exactly one of the sets A and A’ must be in 99. In the case where V is finite, we may strengthen Corollary the next Theorem.
13 and its dual as in
Theorem
15. Assume 1V 114 with V finite. The following are then equivalent: (i) X is closed under both union and intersection. (ii) There exists a consensus family g and subsets A, B of V such that {A, A’} c a and neither B nor B’ is in 97.
Proof.
(ii) a (i). This is an immediate consequence of Lemma 11. (i) a (ii) Using the fact that ( V 12 4 with V finite, and continuing with the notation of the proof of Theorem 12, we may take A = (ol,a2} and B= {a,,03}. Let ~={CE~(V):C>A~~C>A’}.T~~~{A,A’}~~, whileneitherBorB’isinG9, so we need only show that G$is a consensus family. This follows from that fact that for each profile II =(x1,x2, . . . . x,,), 9==[n
{xt:icA}]U[n
{xi:iEA’}].
0
It should be noted that we really need ( V 124 in the above proof. For if 1V ( = 3, then no matter how we define A, either A or A’ will be a singleton. For any choice of B, this singleton will be a member of B or of B’. Closure under unions and intersections also is related to duality. We illustrate this in the next theorem. Theorem
(i) under (ii) under (iii) under
16. Let X be a collection of subsets Every consensus family g on X satisfies union. Every consensus family $8 on X satisfies intersection. Every consensus family 9 on X satisfies union or intersection.
of S. Then 9 c &@* if and only if X is not closed 9* c 95 if and only if X is not closed G@ = ‘9* tf and only if X is not closed
G.D.
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el al. / Neutral
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(iv) Zf X is not closed under intersection or union, then any consensus family
9
or 9* may be regarded as acting on either X or X*. Proof. Since (ii) is dual to (i), and (iii) and (iv) follow immediately from (i) and (ii), we need only consider (i). If G@ C_G8*, then A E 9 G 97* implies A’6 9, so by Theorem 12, X is not closed under unions. The converse implication is obtained by reversing q
these steps.
We close this section by relating the results back to the examples of Section 2. If we let X denote the weak orders of a set with (S, y) as in Example 1, for example, then X is not closed under intersection or under union, so Corollary 14 holds. If X is any finite lattice that is not distributive, and if y(x) is the set of sup-irreducibles under x, then X is not closed under union, so Theorem 12 applies. The situation with n-trees as they are described in Example 3 implies that the n-trees on a set Y are not closed under union, so again Theorem 12 applies.
5. Ordinal considerations
involving
consensus
families
In social choice theory it is highly undesirable to have a neutral consensus method such that S contains a pair of disjoint subsets. Since we have already seen what happens when g contains a set and its complement, there is no loss in generality in applying Theorem 12, and assuming that X is not closed under union. Theorem
17. Let x, y E X, and assume X is not closed under union, and that a contains disjoint subsets A and B. Then x U y E X if and only if x n y E X.
Proof.
Assume
first that xflyczX.
n(a)=x
if acA;
n(a)= y
if aEB;
n(a)=xny
Define
7c as follows:
if aE(AUB)‘.
ThenaEx\yimpliesa,=AE$8; bEy\ximpliesb,=BEED; c,=V~a. It follows that B,=xUy, so xUyeg. Assume next that XU y E X, and define IZ by
z(a)=x
if aeA;
n(a)= y
if aEB;
n(a)=xUy
andccxnyimplies
if aE(AUB)‘.
Then a E x \ y implies a, = A U (A U B)’ = B’ and b E y \x implies 6, = A’. Since X is not closed under union, we know by Theorem 12 that neither A’ nor B’ can be
G.D. Crown et al. / Neutral consensus functions
members of ~3. The only remaining possibility is to consider c, = VEX, whence 6@==xny, and consequently xfly~ Q. 17
241
c EX fly.
Then
Corollary 18. If X is not closed under union, and ifx, y E X can be found such that exactly one of xfl y, XU y is in X, then no consensus family 8 on X can contain a pair of disjoint subsets. Theorem 19. If X is not closed under union, and if x, y E X can be found such that x U y E X, x n y rfX, then any consensus family $3 on X is an order filter containing no disjoint pair of subsets. Proof. The disjointness assertion follows from Corollary
18. By hypothesis, Xis not closed under intersection. To see that 8 is an order filter, let A E 9 with A c B. If B $9, then by Theorem 12*, B’E 9, contrary to the fact that D cannot contain disjoint subsets of S. 0 Conditions similar to those needed for the results on disjointness also produce closure under intersections. The next results serve to illustrate this point. Theorem
20. Assume
that there exist x, y E X such that
(9 xUy@X; (ii) xUycz for some ZEX. (iii) x n y E X. Then any consensus family 68 on X is closed under finite intersections. Proof.
Let A,BEB. Define 71 by n(a)=x for aEA\B; n(a)=y for aeB\A; 7z(a)=z for aEAnB; and n(a)=xny for a@AUB. Then aex\y implies a,=AE9; bEy\x implies b,=BEC@; and cexny implies c,=Y~a. Thus xUy~9~. By (i), xUyc9,,, so there is a d E ~3)~\(xU y). It follows that d, = AnBE%?. 0 Theorem 21. Assume that there exist x, y, z E X such that x U y $ X and x U y c z. Any consensus family g on X which is an order filter of V is then in fact a lattice filter. Proof.
Let A,BE~. Define 7c by n(a)=x for aEA\B; n(a)=y for a$A; and n(a)=z for aEAflB. Then aEx\y implies a,=AEa; bexny implies b,= VE a; and using the fact that 9 is an order filter of V, CE y\x implies that c,=A’U(A~B)=A’UBE%?. Thus xUy~%?~. Since xUyeX, xUycgD,, so 0 there is a dEa.\(xUy). It follows that d,=AnBEO. Corollary 22. Suppose X is not closed under unions or intersections. If either of the
242
C.D. Crown er al. / Neutral consensus functions
following conditions hold, then any monotone-neutral consensus in fact an ultrafilter on V. (i) There exist x, y,zeX such that xUy@X and xUycz. (ii) There exist x, y, z E X such that x Cly $ X and x rl y 1 z. Proof. (i) This follows directly from Corollary is dual to (i). 0
family
9 on X is
14 and the theorem. Condition
(ii)
We did not state a dual to Theorem 21 because we had not yet established the dual of the assertion that $3 be a lattice filter. This is not difficult to do, but we need first to establish some terminology. We agree to call 91 a prime order filter if it is an order filter that satisfies
Thus G@a prime order filter is equivalent to 9’ being a lattice ideal, which in turn is equivalent to g* being a lattice filter. We summarize the situation in the next theorem, and leave to the reader the formulation of the duals to results like Theorem 21. Theorem
23. Let $27be a consensus family on X. Then (i) a is closed under intersection if and only if (%I*)’ is closed under union. (ii) G@’is closed under union if and only if %I* is closed under intersection. (iii) $3 is a lattice filter if and only if kd* is a prime order filter. (iv) 8* is a lattice filter if and only if 9 is a prime order filter. (v) 8 contains no disjoint pair of sets if and only if (91~)’ contains no sets A, B such that A U B= V. (vi) If $27contains no disjoint pair of sets and 9* is a lattice filter, then 8= 8* and $3 is an ultrafilter. Proof. The first five items are routine, and are therefore left to the reader. We concentrate our attention on (vi). By (iv), G@ is a prime order filter. In view of Theorem 9(v), we need only argue that 8 is closed under intersection. So let A, B E 9, and recall that A = (A fl B) U (A fl B’). Since B E a, we must have A fl B’$ $#I. By (iv), 0 %r is a prime order filter. It is immediate that A rl B E 9.
6. A version of Arrow’s
Theorem
The proof of the original version of Arrow’s Theorem (Arrow, 1962) made strong use of the transitivity of weak orders. If X is a set of transitive relations on the set Y, and X is represented as in Example 1, then transitivity may be stated as follows: (x, y) E y(R), (y, z) E y(R) together imply that (x, z) E y(R). The complement of a transitive relation is often said to be negatively transitive. Specifically, to say that
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243
a relation R is negatively transitive (see, for example, the definition of weak order given in Barthelemy and Janowitz, 1991, p. 306) is to say that (x, Y) $ Y(R), (Y,z) $ y(R) together imply that (x,.3 cf y(R). We shall find it more useful to use the contrapositive R is negatively transitive if
of this statement, and say that
(x, z) E y(R) implies that (x, y) E y(R) or (y, z) E y(R). The next definition shows how these concepts can be phrased in terms of the setting provided by Remark 4. Definition 24. Let X be a collection of subsets of S. (i) For kr2, the k-tuple (a,, a2, . . . , ok) of elements of S is said to be transitive in the case where it iS true that for any XEX, {al,a2, ...,ak_l} Sx implieS that akex. The given k-tuple is said to be negatively transitive or dually transitive if ak E x implies that aiEx for some iE (1, . . . . k- 1). (ii) For subsets A,B of S, we say that XE X separates A from B if A c x and B n x = 0. The element y E X is said to include A if A Gy, and the element z E X excludes A if Aflz=O. As indicated in the next lemma, duality plays an important role in connection with the concepts introduced in Definition 24. Lemma 25. (i) The k-tuple (a,, a2, . . . , ak) of &?rTMtS of S is transitive for X if and only if it is dually transitive for X *. (ii) If A, B c S, then XE X separates A from B if and only if X’E X* separates B from A. (iii) If A c S, then XE X includes A if and only if X’E X* excludes A. Proof. Conditions (ii) and (iii) are immediate from the definitions, so we need only establish (i). To do this, we simply note that (a,,az, . . . . ak) transitive 0 for every xcX,
(a,,a2, ...,ak-l>
cx implies akex
0 for every XE X, ak $x implies ai $X for some iC k 0 for every x E X, ak E x’ implies ai E x’ for some i< k *
(al,a2,
. . . . ak)
is dually transitive.
0
We have already noted that Definition 24(i) provides a generalization of transitivity of relations. But in the setting of Example 2 it does much more than that. Suppose, for example, that the finite join-semilattice is represented as families of its set S of sup-irreducibles. Then if a, YES with a>b, the pair (a,b) is transitive. If
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consensus
functions
CZ, b, c E S with c5 aV b, then the triple (a, b, c) is transitive, etc. Even in the context of Example 1, we can vastly generalize the usual notion of transitivity. For example, a triple of ordered pairs {(4 b), (c, d), (x, y)} is transitive with respect to a family X of relations if for any R E X, it is true that aRb and cRd together imply that xRy. When considering the remainder of the paper, the reader should bear in mind the extremely general nature of this notion of transitivity. Until further notice, it is assumed that we are in the setting described in Remark 4. We begin our discussion with a rather interesting connection between transitivity and X being closed under unions or intersections. Lemma
26. For kz3, let (a,, . . . , a,) be a k-tuple of elements of S. (i) Suppose there exist x, y E X and an index i< k - 1 such that x separates from {ak}. If {a ,, . . . . ai> from {ai+ I, . . . . ak} and y separates {ai+l,...,ak_l} ,, . . . , ak) is transitive, then x U y $ X. Hence, for any consensus family a, A ED (a implies A’ $ a. (ii) Suppose there exist x, y E X and an index i< k- 1 such that x separates If {a;, ,, . . . . ak> from {a,,..., a;} and y separates {c(k) from {ai+,,...,ak_,}. (a , , . . . , ak) is negatively transitive, then x fl y $ X. Hence, for any consensus family 97, AeD implies A’E $9. Proof.
(i) If xUyeX, then {a,, . . . . ak_ I } c x U y would imply ak E x U y, a contradiction. (ii) akExny implies that aiExny for some jck, a contradiction. 0
Lemma
27. Suppose that for some k 2 3, S contains a transitive k-tuple (a,, . . . , a&
Suppose further that x, y, z E X can be found so that for some i (1 I i 5 k - 2): x separates(a;+, ,..., ak} from {al ,..., a;},yseparates{a,+, ,..., ak_,} from {al ,..., a;,ak}, and z includes {a,, . . . , ak}. Any consensus family 9 is then an order filter on V. Proof. Let C@ be a consensus family on X. Let A E 8, and A C BC
V. Define a profile rc by rc(a)=z if SEA; n(a)=x if aeB\A; and rr((~)=y if cr@B. Then for a=aj (i+lsjck), aexflyflz, SO a,=V, and aE6@.. If b=aj (lsjci), then By transitivity, it follows that akE 9,. bez\(xUy), so b,=AEa, and bEgz. Hence B = (a& E 8, and this shows 91 to be an order filter of V. 0 The above lemma will be especially useful when k = 3. It should be noted that for a transitive triple (a, b, c), the hypotheses can be stated in a formally weaker manner. It suffices to find x, y,ze X such that x separates {b, c} from a, y separates b from c, and z includes {a, b, c}. The dual of Lemma 27 would assert that if for some k 13, S contains a dually transitive k-tuple (a ,, . . . , ok), and if x’, y’,z’~X* can be found such that for some i (1 ril k-2): X' separates (ai+,, . . . . ak} from {al, . . . . at}, y’ separates {ai+l,...,ak_l} from {al,...,ai,ak}, and z’ includes {a,,...,ak}, then any consensus family 8 (see Theorem 9) is an order filter on V.
G.D.
Crown et al. / Neutral consensus functions
Lemma 28. For kr 3, let (a,, ,.. , ak) be a transitive k-tuple of elements pose there exist g, h, w, z E X and an index i < k - 1 such that g separates from {ai+,,..., ak}, h separates (ai+,, . . . . ak_,} from (a, ,..., ai,ak}, ia 1, .*a,ak>, and w excludes {a,,... , a,(}. Then any consensus family under finite intersections.
245
of S. Sup{a,, . . . , a;) z includes 8 is closed
Proof.LetA,BE%r.Definenby~(cr)=gifaEA\B;7c((r)=hifaEB\A;n(a)=z if aeA tlB; and n(a)= w if cr@A UB. Then for a=aj (1 Qri), a, =A ~62J; for b=aj (i+Icj
Proof. By Lemmas 27 and 28, 9 is a lattice filter. The final assertion follows from Theorem 12*. 0
It is instructive to consider how Theorem 29 relates to the weak orders of a finite set. When represented as in Example 1, Theorem 29 yields the fact that any consensus family is an ultrafilter on V. Indeed, let us consider the transitive triple {(a, b), (b, c), (a,c)}. The required elements g, h, t, w, z can each be taken as a linear order, and it is clearly only necessary to specify their action on the set {a, b, c}. So we take g as c
246
G.D. Crown et al. / Neutral consensus functions
atoms. If each weak order Wis represented as the set of atoms that it contains, then the quota-rule consensus methods (Barthelemy and Janowitz, 1991) are not lattice filters. For example, if rc = (IV,, W,, W,), and if $?8consists of all subsets of { 1,2,3} having at least two members, then
is the median consensus. In fact, there exist consensus families that are not even order filters. To illustrate this, let V= { 1,2,3}, and take F(n) equal to the join of all atoms a such that a, = { 1 } or a, = V. The join exists since we are working in the interval [0, n( 1)], and clearly $8= { ( 1 }, V} is not an order filter of V. Suppose X is either the set of equivalence relations, the set of partial orders, or the set of quasi-orders on some set Y. If X is represented as in Example 1, we leave to the reader the routine verification that the hypotheses of Theorem 29 are satisfied, so that’ this reaffirms the well-known result that every consensus family is in these instances a lattice filter of V. The next lemma is closely related to Lemma 27, and will be useful in Section 7. Lemma
30. Suppose that for some kz 2, S contains a transitive k-tuple (a,, . . . , ak). Suppose further that x, y,z E X can be found such that y separates ak from Any consensus {a ,,..., ak_l}, x includes {a ,,..., ak}, and 2 excludes {al,...,ak}. family g is then an order filter on X. Proof. Let C@ be a consensus family on X. Let A E 9, and A C B C I/. Define a pro-
file II by n(a)rx if aEA; a=a;(lIi
n(a)=y if aeB\A; and n(a)=z if a@B. Then for Hence {at,...,ak_t}Cga,. Bytransitivity, akEgz. It 0
When k=2, the above lemma has some interesting applications. For example, consider the set bof all lattice tolerances on the reals. A typical TE dhere is a reflexive symmetric relation on IRhaving the property that aTb, al crds b implies cTd. Thus the pair ((a, b), (c, d)} is a transitive pair, and Lemma 30 implies that any consensus family on his an order ideal of the underlying set V. A second example can be found in Shmuely (1974). For complete lattices A, B a relation 0 c A x B is called a G-ideal if (Gl) aOb, (c, d) 5 (a, b) together imply c9d, (G2) if q#b, for all a, then (V, a,, A, b,), (A, a,, V, 4-J E 8, (G3) (Rl), (1,0)~8. By Lemma 3.3 and Theorem 1.3 of Shmuely (1974), there is a natural one-to-one correspondence between G-ideals of A x B and galois connections between A and B. Again, Lemma 30 gives us information about the nature of consensus families of G-ideals, and hence about consensus families of galois connections.
G.D. Crown et al. / Neutral consensus functions 7. Applications
247
to finite lattices and semilattices
Throughout this section, X will denote a finite (join or meet) semilattice having more than one member. For each VEX, let J, denote the set of sup-irreducibles under a, and M, the set of meet-irreducibles that do not contain a. Assume for the moment that X is a lattice that is not distributive. Then J,f7 Jb= Johb,, so the representation a- JOleads to a representation that is closed under intersection, but not under unions. For m a meet-irreducible, m L av b if and only if m 2 a or m 2 b, Thus a-M, leads to a representation that is closed under so M,UM,=A4,.~. unions but not intersections. A third representation can be taken by letting a-M,U J,, and this one is not closed under either unions or intersections. Thus the type of consensus method that is possible for a finite semilattice depends very much on the type of building block that is used to represent the lattice. 31. Let X be a finite lattice, and represent X by means of the mapping a-J, as a set of subsets of its sup-irreducibles. Assuming V is finite, every consensus family 8 on X is a lattice jilter if and on/y if X fails to be distributive. Theorem
Note first that if X is distributive, then the representation a++JO is closed under unions and intersections. By Remark lO(iv), every order filter of V is a consensus family. Suppose conversely that X fails to be distributive. The conditions of Theorem 21 trivially hold, so we need only show that a typical consensus family 8 is an order filter. If Xis not atomistic, then there are sup-irreducibles a, b such that b< a. By Lemma 30, $@is an order filter of V. Now assume that X is atomistic. Since X is not distributive, there must be a set U of atoms such that if c = V U, then U{ J, : a E U} C J,. If U= {a,, . . . . ak_ ,}, and if akE J,\ U, then the k-tuple {aI, . . . , ak} is transitive. Note that ak SeparateS ak from {a,, . . . , ak_ 1}, 1 includes (at, . . . , ak} and 0 excludes {a t, . . . , ak}. By Lemma 30, 9 is indeed an order filter of V. 0 Proof.
32. A finite join-semilattice filter of X is a distributive lattice.
Definition
X is said to be distributive if every principal
Theorem 33. Let X be a finite non-distributive join-semilattice
which is not a lattice, and represent X by means of the mapping a-J, as a set of subsets of its supirreducibles. Every consensus family 9 on X is then an ultrafilter. Proof. The representation
of X as subsets of its set S of sup-irreducibles is not closed under unions or under intersections. By Corollary 14, for any subset A of V, exactly one of A and A’ is in 9. By Corollary 18, 91 contains no disjoint pair of subsets. It follows that 9 is an order filter of V. Since X is not distributive, there is an element a E X such that [a, l] is not distributive. Working in that interval, it is easy to produce elements that satisfy the hypotheses of Theorem 21. It follows that g is an ultrafilter of V. 0
248
G.D. Crown et al. / Neutral consensus functions
If X is a distributive finite join-semilattice that is not a lattice, the properties of consensus families on X become much more subtle. First of all, if a 0 element is adjoined to X, then XU (0) becomes a lattice. If that lattice is distributive, then the representation a-J, is closed under unions but not intersections. It follows that there exists a consensus family $9 on X that contains a pair of complementary subsets of V. Further investigation of this will be left for a later paper. On the other hand, if XU (0) is not distributive, then the representation a++J, is not closed under unions or intersections, and much more can be said. Theorem
34. Let X be a finite distributive join-semilattice
that is not a lattice, and represent X as a set of subsets of its sup-irreducibles in the usual manner. If X U { 0) is not distributive, and if X contains at least four atoms, then every consensus family 9 on X is an ultrafilter. Proof. Note first that {Js :gcX}
is not closed under intersection or under union. Hence if 8 is a consensus family, and if A c V, then by Corollary 14, exactly one of A and A’ must be an element of 9. If a consensus family $8 could be found that was not an order filter of V, then there would exist subsets A,B of V such that A c B, A E 97, but Be $21.Hence B’E g, so g contains a pair of disjoint subsets. By hypothesis, X contains distinct atoms a, b,c,d. By Theorem 17, J,U JbC Javb.’ Hence there is an atom c such that clav b, with cc$ {a, b}. By hypothesis, X contains an atom d distinct from {a, b, c}. If d f av b, we have a transitive triple (a, b, c) and an atom d such that (a V b)r\ d = 0. An application of Lemma 30 now contradicts the fact that 9 is not an order filter. Thus we must have dlaV b. But consider the following profile: n(o) = a if a EA; n(a) = b if a E B’; and n(a) = c if (r E B\A. Then both a, and b, are in fa, so a, b E gJR. It follows that d E 93, since dlaV b. But d, = 0, and 0 $9. This final contradiction shows that any consensus family 9J must after all be an order filter. By Corollary 22, any such 91 must be an ultrafilter on v. 0 The reader should note that if X is any finite median semilattice having at least three distinct. atoms a, b, c, then XU (0) is not distributive. For if d is any atom under (avb)A(avc)A(bvc), and dE(a,b,c} we may assume with no loss of generalitythatd=a.Butthend~b~cwithd~{b,c}.Ifd~{a,b,c},weagainhave ds bvc with de {b, c}. In either event, this forces XU (0) to not be distributive. Remark 35. Here is an example that shows that if X has only three atoms, then the conclusion of Theorem 34 need not obtain. Suppose X= {a, b, c, 1 }, where a, b, c are atoms. Let V have three members, and let
~={{1,{2),{3),~1. I Note added in proof: Hence there exists a sup-irreducible SI$ {a, b}. If s>a, we may apply Lemma 30 to the pair (s, a) to deduce that %? is an order filter of K A similar argument applies when s> b. Otherwise, sza
and sz b, so s contains
an atom
c$ {a, b}. The proof
now proceeds
as given in the text.
G.D. Crown et al. / Neutral consensus functions
249
Consider the possible profiles (using symmetry, the following represents every type of profile): n1
=(I, 1,119
7c3 =
(194
a),
712 =(I, 714 =
(La,
776 =
(a,
l,a), 6),
a, b).
Using g as a consensus family, and letting F be the associated consensus method, F(z,)= 1; F(n2)=a since a,=V, b,=c,={1,2}$97; F(ns)=l since ag=V, b,=c,={l}~g; F(n,)=c since a,={1,2}6Q7, b,={1,3}$5Z7, c,={l}~a; F(ns)=l since a,={l}~S?, b,={2}~9, c,={3}~kZI. F(7r6)=b since a,={1,2}BQ!J, b,={3}ES%, c,=O$@. This then produces a consensus family on X that is not an order filter. This leaves open the exact conditions under which Theorem 34 remains true when X has fewer than four atoms.
8. Conclusion We have presented a new model for those neutral consensus methods that are defined in terms of stability families. Included in this model is a notion of duality that plays a key role in understanding the connection between closure under unions and closure under intersections. Our goal was to investigate the properties of stability families that serve to guarantee that every neutral consensus family D be an order filter, be closed under intersections, be a lattice filter, to not have pairs of disjoint sets, and to either contain or fail to contain a pair of complementary subsets of V. This is a project that is still under active investigation. We have demonstrated a clear connection between these ideas, and a generalization of transitivity of relations, and its connection with an abstract version of Arrow’s Theorem. Our results include as special cases much of the standard theory of neutral consensus methods, and thus serve as a natural mathematical model for their study. It has the advantage of presenting the subject in a way that discounts the nature of the objects themselves, but concentrates instead on the manner in which the building blocks in the stability families are put together to form a consensus.
Acknowledgment
The second author’s 90-1008.
research was supported
in part by ONR Grant NOOO14-
G.D. Crown et al. / Neutral consensus functions
250
References K.P. Arrow, J.-P.
Math. J.-P.
Social Choice
Barthelemy Sot.
Sci.
Barthelemy
305-322. P. Buneman,
and Individual
and B. Monjardet,
2nd edn. (Wiley, procedure
New York,
in cluster
analysis
of consensus,
SIAM
The recovery
Janowitz,
Edinburgh,
1971), 387-395. Arrowian
A formal
of trees from measures
eds., Mathematics
in Archaeological
characterizations
theory
of dissimilarity, and Historical
of latticial
federation
On galois
connections,
Pacific
J. Math.
J. Discr.
in: F.R. Hodgson, Sciences
(Edinburgh
consensus
20 (1990) 51-71. 2. Shmuely,
1962) (1st edn.,
1951).
and social choice theory,
1 (1981) 235-268. and M.F.
P. Tautu, B. Monjardet,
Values,
The median
54 (2) (1974) 209-225.
functions,
Math.
4 (1991)
D.G. Kendall University Math.
and
Press, Sot.
Sci.
Social
Sciences
231
25 (1993) 231-250
North-Holland
Neutral
consensus
functions
Gary D. Crown Wichita State University, Wichila, KS 67208, USA
Melvin F. Janowitz Universiry of Massachusetts, Amherst, MA 01003. USA
Robert C. Powers University of Louisville, Louisville, KY 40292, USA Communicated
by F.W. Roush
Received August
1992
Neutral consensus methods have been studied in detail by a number of authors in specific contexts. This paper introduces a general mathematical model that allows for a unified approach to their study via the notion of stability families. Special attention Key words: Consensus
method;
is paid to conditions
neutral;
that produce
abstract
versions of Arrow’s Theorem.
stability family.
1. Introduction The need to summarize a collection of structures into a single object has arisen in such diverse areas as social choice theory, evolutionary biology, numerical taxonomy, economics, psychology and classical statistics. Such a procedure is often called a consensus technique or a consensus method. It has long been recognized that because consensus theory has independently been developed in many diverse disciplines, there is a need for a general mathematical model that will produce results having wide-ranging interpretations. One such approach is by means of ordered sets. The present work has its genesis in two recent approaches to these issues by means of various ordinal structures (see Barthelemy and Janowitz, 1991, as well as Monjardet, 1990). For further background the reader might consult the references mentioned in these sources. It is worthy of mention though that much Correspondence to: M.F. Janowitz,
Department
of Mathematics,
University
of Massachusetts,
MA 01003, USA.
01654896/93/$06.00
0
1993-Elsevier
Science
Publishers
B.V. All rights
reserved
Amherst,
232
G.D.
Crown
et al. / Neutral
consensus
functions
of the activity in consensus theory stems from Arrow’s seminal work (Arrow, 1962). Essentially Arrow produced a number of seemingly reasonable axioms that a consensus method on weak orders should satisfy, and then proved his celebrated ‘impossibility’ theorem - that no such consensus method could exist. Since then much research has been devoted to extending Arrow’s Theorem to other contexts, and seeking ways to modify the axioms underlying Arrow’s Theorem so as to avoid its rather distressing consequences. Our general approach will be to use an idea of Barthelemy and Monjardet (1981), and consider the construction of a consensus object as a process where certain ‘building blocks’ or ‘bricks’ are glued together in a suitable manner. Although we shall keep the basic setting arbitrary, we will when appropriate show how our results generalize some of the standard theory on consensus. A major goal will be to determine just what it is about the ‘building blocks’ that produces
analogs
of Arrow’s
Theorem.
2. Notation The basic notation and terminology of the paper will be established here. Before one can even think of performing a consensus, one must identify the set of objects to be summarized. We agree to let X denote this set. Although some of the relevant examples are finite, we impose no such restriction on X; indeed, we agree in general that X shall denote a possibly infinite set having at least three members. In social choice theory, X might represent subsets of the set of alternatives under consideration; in voting theory, X might denote the linear orders or the weak orders on the set of candidates, etc. V will denote a fixed (possibly infinite) indexing set having at least two members. When V is finite, it will be useful to let V= { 1,2, . . . , n} for some positive integer n > 1. Think of V as the set of voters or the set of judges or the set of committee members who are responsible for producing the consensus object. P= XV will denote the set of mappings from V into X. A member of P is called a profile. In voting theory, one can think of an element of P as the collection of ballots in an ‘election. For V finite, a typical profile will be 7~= (x1,x2, . . . ,x,) with xi E X for all i. In the infinite case, one can think of a profile as an indexed family where rc((r)=x, for each a~ Y. 7r =(x,),, y or else as a function 7c: V/-X, A consensus function or a consensus method may now be taken as a mapping F: P - X. Other authors who have considered consensus methods have allowed the set V to vary, and have examined the effects of voters being added or deleted from I’. They have also considered consensus functions that produce non-empty subsets of X rather than single elements, or even take values in some set other than X. Issues such as these will be considered in a later paper. Finally (as mentioned earlier), we introduce a set S of building blocks or bricks that can be used to construct the output consensus object (see Barthelemy and Monjardet, 1981). The general approach will be to study consensus methods in terms of
G.D. Crown et al. / Neutral consensus funcrions
233
how they interact with these building blocks. Thus a stability family on Xis defined to be a pair (S, y), where S is a given set of building blocks and y :X - P(S) is a mapping. Here are some examples that the reader should keep in mind. 1: Binary relations. Let Y be an underlying set, and X some set of binary relations on Y. Let S = Y x Y, and for each R EX, let y(R) = {(x, y) : xRy} . So the idea here is to form a consensus relation F(rc) from a profile 7~by determining the pairs (x, y) that would belong to the consensus. Of course the problem is compounded by the need to have F(n) E X. Alternatively, one could let y(R) = {(x, y) : xRy fails}. Example
Example 2: Finite lattices and semilattices. Let X be a finite lattice. Then every element of X is a join of sup-irreducibles and a meet of inf-irreducibles. Thus we can take S = either the set of sup-irreducibles or the set of inf-irreducibles of X, and in the first instance define
y,(x)=(bES:blx}; in the second instance, take yz(x)={bES:bzx}. We could also use the dual of either of these examples, or even take S = (s E X: s is sup-irreducible or inf-irreducible), with v(x) = y,(x) U ya(x). Note here that we do not regard the smallest element of a lattice to be sup-irreducible, since 0= V {xEX:X
of Y satisfying: l
YET.
l
0 is not in T.
l l
{y}ETfor
aIlyEY.
A,BE T implies A nBE {O,A,B}.
The members of T are called clusters, and the members other than X and the singletons are the proper clusters. We can now take S = B(Y), and for each n-tree T, let y(T) denote its proper clusters. Thus a consensus method F would construct F(n) by choosing its proper clusters. Similar examples may be formed by consider-
G.D. Crown el al. / Neutral consensus fundons
234
ing other types of families (Buneman, 1971).
3. Consensus
families
of subsets of a set. Among
and neutral consensus
them are thephy/ogenetic
trees
functions
Let F be a consensus function, (S, y) a fixed stability with the notation introduced in Section 2.
family
on X and continue
Definition 1. (i) For each XEX, let 7cXdenote a profile whose entries are all X. (ii) F is called neutral if for all profiles rc and rc’ and for all s, S’E S, it is true that if {BE v:sE)+r(a))}={crE
V:s’E)@‘(a))$,
then SE y(F(n))
if and only if S’E y(F(n’)).
(iii) F is called monotone-neutral it is true that if
if for all profiles
71 and 71’ and for all S,S’E S,
then s E y(F(~c)) implies
s’ E y(F(n’)).
Lemma 2 contains a useful and well-known fact about neutral and monotoneneutral consensus functions that will provide a central theme for the remainder of the discussion. 1991, Lemma 3 and 4). The consensus 2. (Barthtlemy and Janowitz, method F is neutral of and only if there is a collection C@)F of subsets of V having the property that
Lemma
s~y(F(z))
if and only if {a:sEy(x(a))}
It is monotone-neutral if it is neutral and the family sense that A E aF, A C_B implies BE BF.
E !SF. kDF is an order filter in the
Order filters are called federations in Monjardet (1990). The idea now is to relate properties of y with the nature of the possible collections gF of subsets of V that are associated with neutral consensus functions. The start of this work is in Monjardet (1990), and it played a role in Barthelemy and Janowitz (1991) in the context of Example 2. We would like here to view the material in as genera1 a setting as possible, and (when feasible) in each instance to obtain necessary and sufficient conditions for a given property to hold.
Remark
3. It seems reasonable
to require
that
S= UxeX v(x). This
follows
from
G.D. Crown et al. / Neutral consensus functions
235
the fact that if se y(x) for any XE X, then s is not going to have any effect on a consensus function F. If y is not one-to-one, then it can only determine a neutral consensus function up to the equivalence relation induced by y. In this case it seems appropriate to just idenfity x withy in the case y(x) = y(y), and work with the quotient set obtained by this identification. Once this is done, y induces a one-to-one mapping on this set. In view of this, we shall henceforth simply assume that for any stability famiiy (S, y), the mapping y is one-to-one. Remark 4. (i) The notation
will be greatly simplified, and many arguments will become easier to understand if we simply identify each XE X with the corresponding subset y(x) of S. Accordingly, we herewith change our setting: let S be a set having at least three members, and XC 9(S), leaving the terminology involving V unchanged. (ii) For a E S and II E P, it will be convenient to let a, = {a E V: a E n(a)}. (iii) Let $2 be a family of non-empty subsets of Vsuch that VE 97. For each rc E P, we let 9, = {a E S : a, E 95}. Finally, we may define a relation 0 from S to P by the rule aOx if and only if a, E 9, and note that 0 induces a galois connection between B(S) and 9(P) by the rules: a(T)={7rEP:a&r
for all aET},
jl(Q)={aeS:aOn
for all ZEQ},
for all subsets Tc S and Q c P. We assume that the reader has a basic knowledge of the theory of galois connections. The results we shall use can be found in any standard text on lattices and ordered sets. In particular, the subsets Tc S and Q c P will be called closed in the case
T = PM”)), Lemma Proof.
Q = d.NQ)).
5. For each x E X, x is closed.
We simply note that
B(dx)) =PW%J
= B(dD({n,)))) =P({7c,)) =x.
Definition 6. The set 91 of subsets of V is
~27as a neutral
n,n(a) c gd, c n, n(a).
0
G.D. Crown el al. / Neutral consensus functions
236
Before proceeding, we repeat the connection between the two approaches to neutral consensus functions. On the one hand, we have a set X and a stability family (S, y). By identifying each element of X with its image under y, we produce a family of subsets of S that induces a neutral consensus method in the sense of Definition 6. Suppose, on the other hand, that $27is a consensus family in the sense of Definition 6. Then for each rc E P, ~2)~E X. We may define a mapping F: P - X by the rule F(x) = 9),, take v(x) =x for all XE X, and note that (S, y) is a stability family for F and that F becomes a neutral consensus function in the usual sense, in that s E F(x) if and only if s, E %,. Remark 7. We agree to order X by set inclusion on S. If X is closed under unions,
then it clearly forms a join-semilattice; similarly, closure under intersection makes it into a meet-semilattice. If it is closed under both unions and intersections, then it is a distributive lattice. It is important to notice that X can be a distributive lattice without it being closed under both unions and intersections. To illustrate this concretely, we ask the reader to recall that a weak order on a set Y is a binary relation W that is reflexive, transitive and complete in the sense that for all x, y E Wit is true that x Wy or y Wx. Consider the set X of weak orders on a set Y having at least four elements, taking as a stability family Y x Y with y(W) = {(x, y) : x Wy} for each weak order W. When ordered by inclusion, X forms a join-semilattice that is not a lattice, but which has the property that each of its principal filters is a distributive lattice. Since the union of a pair of weak orders need not be transitive, it is not true that y(W,VW2)=y(Wl)Uy(W2) for all W,, W,EX. Definition 8. There is an interesting notion of duality related to all of this. Let X be a family of subsets of S with V defined in the usual manner. Let
X*={X’:XE~),
where x’=S\x,
p*=(X*)&{n*:
n E P} , where n *(cr) = [n(o)]‘.
If ‘3 is a consensus family on X, define the dual family 9*=(/I
ka* by
c V:A’$kD}.
It will also be convenient to define the complementary family of 8 by the formula a’=&4
c I’:A$G!r}.
Note that the mapping A - A’ is a dual automorphism of 9(V) that maps 91’ onto a*, and G@onto (g*)‘. The next theorem gathers up the salient facts about this notion of duality. 9. Let 43 be a consensus family on X. Then: (i) 52J*is a consensus family on X*. (ii) For each 7c* E P*, a,*, = (~3~)‘. (iii) 5%= a**.
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(iv) 9 is monotone-neutral if and only if $B* is monotone-neutral. (v) $9 is an ultrafilter if and only if LB* is an ultrafilter. In fact, if this is true, then G3= L@*. (iii) will each follow directly from (ii), so we concentrate on (ii). We simply note here that for each a ES,
Proof.
our efforts
This shows that aE $B,*,*ae8,, thus establishing (ii). (iv) This is an immediate consequence of the fact that GZJis an order filter if and only if $2~‘isan order ideal, and this in turn is equivalent to GB*being an order filter. (v) To say that 9 is an ultrafilter is equivalent to saying that &&is a maximal lattice ideal, and this in turn is equivalent to saying that 8* is an ultrafilter. To complete the proof, note that if 9 is an ultrafilter, then AEGB iffA’$%7 iffAED*.
0
Remark 10. (i) When X is either closed under unions or intersections,
a number of concrete examples may be constructed. Here is the underlying principle. Let $3, and $_Qbe consensus families on X. Let g3 = 6BD1 U C@*,and g4 = 9, fl D,. If X is closed under union, then G@J~ is a consensus family and for any profile 71, 5B3,= * similarly, if X is closed under intersection, then &B4is a consensus g1, u a,, family with G2J4, = G3’r.fl $B2,. Since X is closed under union if and only if X* is closed under intersection, these two constructions are dual to each other under the construction outlined in Theorem 9. (ii) Suppose X is closed under unions, and let A be a finite nonempty subset of V. Then 22l={C~ V:aEC for some crEA} is a consensus family, and for each profile z, we have
can= u
7c(a).
asA
(iii) Suppose X is closed under intersections, a=(Cr
with A as in (ii). This time, let
V:AcC}.
and note that a,=
n
77(a).
CIEA
(iv) Finally, assume X is closed under both unions and intersections. Let V be finite, and assume $2 is an order filter of V. Then $8 is a consensus family on X.
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If A,,A*,
. ..) Ak are the minimal
elements
of $3, then
for any profile
rc,
4. Complements We continue with the notation introduced in Remark 4, and consider the question of complementary subsets of I/ either being or not being members of a consensus family G@. Lemma 11. Let %! be a consensus family on X. (i) If there exists a subset A of V such that both A and A’ are in $3, then X is
closed under unions. (ii) If there exists a subset A of V such that neither A nor A’ are in 525,then X is closed under intersections. Proof.
(i) Let x, y E X, and define
z(a)=x
for aeA
n(a)= y
for aEA’.
a profile
rc by requiring
and
IfaEx\y,thena,=AE4);similarly,aEy\ximpliesa,=A’E~; forces a,=VECB. Since ~J~CXUY, it follows (ii) This is dual to (i). Cl
finally,aExny that
g,=xUy~X.
The next two theorems further develop this natural connection between complementary subsets of V and whether or not X is closed under unions or intersections. Theorem
A’$8
12. Every consensus family 9 on X has the property that A E 9 implies if and only if X is not closed under unions.
Proof. Assume the existence of x, y E X such that XU y 13X. By the contrapositive to Lemma 1 l(i), A E $2~implies A’b 9~. Suppose on the other hand that X is closed under union. Let A c V be non-empty. Choose fixed elements a E A, a’E A’, and let $2 be the collection of subsets of V that contain a or a’. Notice that both A and A’ are members of 93, so we need only show that 23 is a consensus family. But this 0 is easy to do. For any profile rr, we simply note that gD, = n(a) U n(a’) E X. Corollary 13. There exists a consensus family C?ZJ on X that contains complementary subsets of V if and only if X is closed under unions.
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The next results follow by applying Theorem 12 to X*. Hence they are dual to Theorem 12 and Corollary 13. Theorem
12*. Every consensus family 91 has the property that for every subset B implies B' E 9 if and only if X is not closed under intersection.
of V, B$9 Corollary
13*. There exists a consensus family 9 on X whose complement contains a complementary pair of subsets of V if and only if X is closed under intersections. Corollary 14. Assume X is not closed under intersection or union. Then for any con-
sensus family Q7on X, and for any subset A of V, exactly one of the sets A and A’ must be in 99. In the case where V is finite, we may strengthen Corollary the next Theorem.
13 and its dual as in
Theorem
15. Assume 1V 114 with V finite. The following are then equivalent: (i) X is closed under both union and intersection. (ii) There exists a consensus family g and subsets A, B of V such that {A, A’} c a and neither B nor B’ is in 97.
Proof.
(ii) a (i). This is an immediate consequence of Lemma 11. (i) a (ii) Using the fact that ( V 12 4 with V finite, and continuing with the notation of the proof of Theorem 12, we may take A = (ol,a2} and B= {a,,03}. Let ~={CE~(V):C>A~~C>A’}.T~~~{A,A’}~~, whileneitherBorB’isinG9, so we need only show that G$is a consensus family. This follows from that fact that for each profile II =(x1,x2, . . . . x,,), 9==[n
{xt:icA}]U[n
{xi:iEA’}].
0
It should be noted that we really need ( V 124 in the above proof. For if 1V ( = 3, then no matter how we define A, either A or A’ will be a singleton. For any choice of B, this singleton will be a member of B or of B’. Closure under unions and intersections also is related to duality. We illustrate this in the next theorem. Theorem
(i) under (ii) under (iii) under
16. Let X be a collection of subsets Every consensus family g on X satisfies union. Every consensus family $8 on X satisfies intersection. Every consensus family 9 on X satisfies union or intersection.
of S. Then 9 c &@* if and only if X is not closed 9* c 95 if and only if X is not closed G@ = ‘9* tf and only if X is not closed
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(iv) Zf X is not closed under intersection or union, then any consensus family
9
or 9* may be regarded as acting on either X or X*. Proof. Since (ii) is dual to (i), and (iii) and (iv) follow immediately from (i) and (ii), we need only consider (i). If G@ C_G8*, then A E 9 G 97* implies A’6 9, so by Theorem 12, X is not closed under unions. The converse implication is obtained by reversing q
these steps.
We close this section by relating the results back to the examples of Section 2. If we let X denote the weak orders of a set with (S, y) as in Example 1, for example, then X is not closed under intersection or under union, so Corollary 14 holds. If X is any finite lattice that is not distributive, and if y(x) is the set of sup-irreducibles under x, then X is not closed under union, so Theorem 12 applies. The situation with n-trees as they are described in Example 3 implies that the n-trees on a set Y are not closed under union, so again Theorem 12 applies.
5. Ordinal considerations
involving
consensus
families
In social choice theory it is highly undesirable to have a neutral consensus method such that S contains a pair of disjoint subsets. Since we have already seen what happens when g contains a set and its complement, there is no loss in generality in applying Theorem 12, and assuming that X is not closed under union. Theorem
17. Let x, y E X, and assume X is not closed under union, and that a contains disjoint subsets A and B. Then x U y E X if and only if x n y E X.
Proof.
Assume
first that xflyczX.
n(a)=x
if acA;
n(a)= y
if aEB;
n(a)=xny
Define
7c as follows:
if aE(AUB)‘.
ThenaEx\yimpliesa,=AE$8; bEy\ximpliesb,=BEED; c,=V~a. It follows that B,=xUy, so xUyeg. Assume next that XU y E X, and define IZ by
z(a)=x
if aeA;
n(a)= y
if aEB;
n(a)=xUy
andccxnyimplies
if aE(AUB)‘.
Then a E x \ y implies a, = A U (A U B)’ = B’ and b E y \x implies 6, = A’. Since X is not closed under union, we know by Theorem 12 that neither A’ nor B’ can be
G.D. Crown et al. / Neutral consensus functions
members of ~3. The only remaining possibility is to consider c, = VEX, whence 6@==xny, and consequently xfly~ Q. 17
241
c EX fly.
Then
Corollary 18. If X is not closed under union, and ifx, y E X can be found such that exactly one of xfl y, XU y is in X, then no consensus family 8 on X can contain a pair of disjoint subsets. Theorem 19. If X is not closed under union, and if x, y E X can be found such that x U y E X, x n y rfX, then any consensus family $3 on X is an order filter containing no disjoint pair of subsets. Proof. The disjointness assertion follows from Corollary
18. By hypothesis, Xis not closed under intersection. To see that 8 is an order filter, let A E 9 with A c B. If B $9, then by Theorem 12*, B’E 9, contrary to the fact that D cannot contain disjoint subsets of S. 0 Conditions similar to those needed for the results on disjointness also produce closure under intersections. The next results serve to illustrate this point. Theorem
20. Assume
that there exist x, y E X such that
(9 xUy@X; (ii) xUycz for some ZEX. (iii) x n y E X. Then any consensus family 68 on X is closed under finite intersections. Proof.
Let A,BEB. Define 71 by n(a)=x for aEA\B; n(a)=y for aeB\A; 7z(a)=z for aEAnB; and n(a)=xny for a@AUB. Then aex\y implies a,=AE9; bEy\x implies b,=BEC@; and cexny implies c,=Y~a. Thus xUy~9~. By (i), xUyc9,,, so there is a d E ~3)~\(xU y). It follows that d, = AnBE%?. 0 Theorem 21. Assume that there exist x, y, z E X such that x U y $ X and x U y c z. Any consensus family g on X which is an order filter of V is then in fact a lattice filter. Proof.
Let A,BE~. Define 7c by n(a)=x for aEA\B; n(a)=y for a$A; and n(a)=z for aEAflB. Then aEx\y implies a,=AEa; bexny implies b,= VE a; and using the fact that 9 is an order filter of V, CE y\x implies that c,=A’U(A~B)=A’UBE%?. Thus xUy~%?~. Since xUyeX, xUycgD,, so 0 there is a dEa.\(xUy). It follows that d,=AnBEO. Corollary 22. Suppose X is not closed under unions or intersections. If either of the
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following conditions hold, then any monotone-neutral consensus in fact an ultrafilter on V. (i) There exist x, y,zeX such that xUy@X and xUycz. (ii) There exist x, y, z E X such that x Cly $ X and x rl y 1 z. Proof. (i) This follows directly from Corollary is dual to (i). 0
family
9 on X is
14 and the theorem. Condition
(ii)
We did not state a dual to Theorem 21 because we had not yet established the dual of the assertion that $3 be a lattice filter. This is not difficult to do, but we need first to establish some terminology. We agree to call 91 a prime order filter if it is an order filter that satisfies
Thus G@a prime order filter is equivalent to 9’ being a lattice ideal, which in turn is equivalent to g* being a lattice filter. We summarize the situation in the next theorem, and leave to the reader the formulation of the duals to results like Theorem 21. Theorem
23. Let $27be a consensus family on X. Then (i) a is closed under intersection if and only if (%I*)’ is closed under union. (ii) G@’is closed under union if and only if %I* is closed under intersection. (iii) $3 is a lattice filter if and only if kd* is a prime order filter. (iv) 8* is a lattice filter if and only if 9 is a prime order filter. (v) 8 contains no disjoint pair of sets if and only if (91~)’ contains no sets A, B such that A U B= V. (vi) If $27contains no disjoint pair of sets and 9* is a lattice filter, then 8= 8* and $3 is an ultrafilter. Proof. The first five items are routine, and are therefore left to the reader. We concentrate our attention on (vi). By (iv), G@ is a prime order filter. In view of Theorem 9(v), we need only argue that 8 is closed under intersection. So let A, B E 9, and recall that A = (A fl B) U (A fl B’). Since B E a, we must have A fl B’$ $#I. By (iv), 0 %r is a prime order filter. It is immediate that A rl B E 9.
6. A version of Arrow’s
Theorem
The proof of the original version of Arrow’s Theorem (Arrow, 1962) made strong use of the transitivity of weak orders. If X is a set of transitive relations on the set Y, and X is represented as in Example 1, then transitivity may be stated as follows: (x, y) E y(R), (y, z) E y(R) together imply that (x, z) E y(R). The complement of a transitive relation is often said to be negatively transitive. Specifically, to say that
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243
a relation R is negatively transitive (see, for example, the definition of weak order given in Barthelemy and Janowitz, 1991, p. 306) is to say that (x, Y) $ Y(R), (Y,z) $ y(R) together imply that (x,.3 cf y(R). We shall find it more useful to use the contrapositive R is negatively transitive if
of this statement, and say that
(x, z) E y(R) implies that (x, y) E y(R) or (y, z) E y(R). The next definition shows how these concepts can be phrased in terms of the setting provided by Remark 4. Definition 24. Let X be a collection of subsets of S. (i) For kr2, the k-tuple (a,, a2, . . . , ok) of elements of S is said to be transitive in the case where it iS true that for any XEX, {al,a2, ...,ak_l} Sx implieS that akex. The given k-tuple is said to be negatively transitive or dually transitive if ak E x implies that aiEx for some iE (1, . . . . k- 1). (ii) For subsets A,B of S, we say that XE X separates A from B if A c x and B n x = 0. The element y E X is said to include A if A Gy, and the element z E X excludes A if Aflz=O. As indicated in the next lemma, duality plays an important role in connection with the concepts introduced in Definition 24. Lemma 25. (i) The k-tuple (a,, a2, . . . , ak) of &?rTMtS of S is transitive for X if and only if it is dually transitive for X *. (ii) If A, B c S, then XE X separates A from B if and only if X’E X* separates B from A. (iii) If A c S, then XE X includes A if and only if X’E X* excludes A. Proof. Conditions (ii) and (iii) are immediate from the definitions, so we need only establish (i). To do this, we simply note that (a,,az, . . . . ak) transitive 0 for every xcX,
(a,,a2, ...,ak-l>
cx implies akex
0 for every XE X, ak $x implies ai $X for some iC k 0 for every x E X, ak E x’ implies ai E x’ for some i< k *
(al,a2,
. . . . ak)
is dually transitive.
0
We have already noted that Definition 24(i) provides a generalization of transitivity of relations. But in the setting of Example 2 it does much more than that. Suppose, for example, that the finite join-semilattice is represented as families of its set S of sup-irreducibles. Then if a, YES with a>b, the pair (a,b) is transitive. If
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CZ, b, c E S with c5 aV b, then the triple (a, b, c) is transitive, etc. Even in the context of Example 1, we can vastly generalize the usual notion of transitivity. For example, a triple of ordered pairs {(4 b), (c, d), (x, y)} is transitive with respect to a family X of relations if for any R E X, it is true that aRb and cRd together imply that xRy. When considering the remainder of the paper, the reader should bear in mind the extremely general nature of this notion of transitivity. Until further notice, it is assumed that we are in the setting described in Remark 4. We begin our discussion with a rather interesting connection between transitivity and X being closed under unions or intersections. Lemma
26. For kz3, let (a,, . . . , a,) be a k-tuple of elements of S. (i) Suppose there exist x, y E X and an index i< k - 1 such that x separates from {ak}. If {a ,, . . . . ai> from {ai+ I, . . . . ak} and y separates {ai+l,...,ak_l} ,, . . . , ak) is transitive, then x U y $ X. Hence, for any consensus family a, A ED (a implies A’ $ a. (ii) Suppose there exist x, y E X and an index i< k- 1 such that x separates If {a;, ,, . . . . ak> from {a,,..., a;} and y separates {c(k) from {ai+,,...,ak_,}. (a , , . . . , ak) is negatively transitive, then x fl y $ X. Hence, for any consensus family 97, AeD implies A’E $9. Proof.
(i) If xUyeX, then {a,, . . . . ak_ I } c x U y would imply ak E x U y, a contradiction. (ii) akExny implies that aiExny for some jck, a contradiction. 0
Lemma
27. Suppose that for some k 2 3, S contains a transitive k-tuple (a,, . . . , a&
Suppose further that x, y, z E X can be found so that for some i (1 I i 5 k - 2): x separates(a;+, ,..., ak} from {al ,..., a;},yseparates{a,+, ,..., ak_,} from {al ,..., a;,ak}, and z includes {a,, . . . , ak}. Any consensus family 9 is then an order filter on V. Proof. Let C@ be a consensus family on X. Let A E 8, and A C BC
V. Define a profile rc by rc(a)=z if SEA; n(a)=x if aeB\A; and rr((~)=y if cr@B. Then for a=aj (i+lsjck), aexflyflz, SO a,=V, and aE6@.. If b=aj (lsjci), then By transitivity, it follows that akE 9,. bez\(xUy), so b,=AEa, and bEgz. Hence B = (a& E 8, and this shows 91 to be an order filter of V. 0 The above lemma will be especially useful when k = 3. It should be noted that for a transitive triple (a, b, c), the hypotheses can be stated in a formally weaker manner. It suffices to find x, y,ze X such that x separates {b, c} from a, y separates b from c, and z includes {a, b, c}. The dual of Lemma 27 would assert that if for some k 13, S contains a dually transitive k-tuple (a ,, . . . , ok), and if x’, y’,z’~X* can be found such that for some i (1 ril k-2): X' separates (ai+,, . . . . ak} from {al, . . . . at}, y’ separates {ai+l,...,ak_l} from {al,...,ai,ak}, and z’ includes {a,,...,ak}, then any consensus family 8 (see Theorem 9) is an order filter on V.
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Lemma 28. For kr 3, let (a,, ,.. , ak) be a transitive k-tuple of elements pose there exist g, h, w, z E X and an index i < k - 1 such that g separates from {ai+,,..., ak}, h separates (ai+,, . . . . ak_,} from (a, ,..., ai,ak}, ia 1, .*a,ak>, and w excludes {a,,... , a,(}. Then any consensus family under finite intersections.
245
of S. Sup{a,, . . . , a;) z includes 8 is closed
Proof.LetA,BE%r.Definenby~(cr)=gifaEA\B;7c((r)=hifaEB\A;n(a)=z if aeA tlB; and n(a)= w if cr@A UB. Then for a=aj (1 Qri), a, =A ~62J; for b=aj (i+Icj
Proof. By Lemmas 27 and 28, 9 is a lattice filter. The final assertion follows from Theorem 12*. 0
It is instructive to consider how Theorem 29 relates to the weak orders of a finite set. When represented as in Example 1, Theorem 29 yields the fact that any consensus family is an ultrafilter on V. Indeed, let us consider the transitive triple {(a, b), (b, c), (a,c)}. The required elements g, h, t, w, z can each be taken as a linear order, and it is clearly only necessary to specify their action on the set {a, b, c}. So we take g as c
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atoms. If each weak order Wis represented as the set of atoms that it contains, then the quota-rule consensus methods (Barthelemy and Janowitz, 1991) are not lattice filters. For example, if rc = (IV,, W,, W,), and if $?8consists of all subsets of { 1,2,3} having at least two members, then
is the median consensus. In fact, there exist consensus families that are not even order filters. To illustrate this, let V= { 1,2,3}, and take F(n) equal to the join of all atoms a such that a, = { 1 } or a, = V. The join exists since we are working in the interval [0, n( 1)], and clearly $8= { ( 1 }, V} is not an order filter of V. Suppose X is either the set of equivalence relations, the set of partial orders, or the set of quasi-orders on some set Y. If X is represented as in Example 1, we leave to the reader the routine verification that the hypotheses of Theorem 29 are satisfied, so that’ this reaffirms the well-known result that every consensus family is in these instances a lattice filter of V. The next lemma is closely related to Lemma 27, and will be useful in Section 7. Lemma
30. Suppose that for some kz 2, S contains a transitive k-tuple (a,, . . . , ak). Suppose further that x, y,z E X can be found such that y separates ak from Any consensus {a ,,..., ak_l}, x includes {a ,,..., ak}, and 2 excludes {al,...,ak}. family g is then an order filter on X. Proof. Let C@ be a consensus family on X. Let A E 9, and A C B C I/. Define a pro-
file II by n(a)rx if aEA; a=a;(lIi
n(a)=y if aeB\A; and n(a)=z if a@B. Then for Hence {at,...,ak_t}Cga,. Bytransitivity, akEgz. It 0
When k=2, the above lemma has some interesting applications. For example, consider the set bof all lattice tolerances on the reals. A typical TE dhere is a reflexive symmetric relation on IRhaving the property that aTb, al crds b implies cTd. Thus the pair ((a, b), (c, d)} is a transitive pair, and Lemma 30 implies that any consensus family on his an order ideal of the underlying set V. A second example can be found in Shmuely (1974). For complete lattices A, B a relation 0 c A x B is called a G-ideal if (Gl) aOb, (c, d) 5 (a, b) together imply c9d, (G2) if q#b, for all a, then (V, a,, A, b,), (A, a,, V, 4-J E 8, (G3) (Rl), (1,0)~8. By Lemma 3.3 and Theorem 1.3 of Shmuely (1974), there is a natural one-to-one correspondence between G-ideals of A x B and galois connections between A and B. Again, Lemma 30 gives us information about the nature of consensus families of G-ideals, and hence about consensus families of galois connections.
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247
to finite lattices and semilattices
Throughout this section, X will denote a finite (join or meet) semilattice having more than one member. For each VEX, let J, denote the set of sup-irreducibles under a, and M, the set of meet-irreducibles that do not contain a. Assume for the moment that X is a lattice that is not distributive. Then J,f7 Jb= Johb,, so the representation a- JOleads to a representation that is closed under intersection, but not under unions. For m a meet-irreducible, m L av b if and only if m 2 a or m 2 b, Thus a-M, leads to a representation that is closed under so M,UM,=A4,.~. unions but not intersections. A third representation can be taken by letting a-M,U J,, and this one is not closed under either unions or intersections. Thus the type of consensus method that is possible for a finite semilattice depends very much on the type of building block that is used to represent the lattice. 31. Let X be a finite lattice, and represent X by means of the mapping a-J, as a set of subsets of its sup-irreducibles. Assuming V is finite, every consensus family 8 on X is a lattice jilter if and on/y if X fails to be distributive. Theorem
Note first that if X is distributive, then the representation a++JO is closed under unions and intersections. By Remark lO(iv), every order filter of V is a consensus family. Suppose conversely that X fails to be distributive. The conditions of Theorem 21 trivially hold, so we need only show that a typical consensus family 8 is an order filter. If Xis not atomistic, then there are sup-irreducibles a, b such that b< a. By Lemma 30, $@is an order filter of V. Now assume that X is atomistic. Since X is not distributive, there must be a set U of atoms such that if c = V U, then U{ J, : a E U} C J,. If U= {a,, . . . . ak_ ,}, and if akE J,\ U, then the k-tuple {aI, . . . , ak} is transitive. Note that ak SeparateS ak from {a,, . . . , ak_ 1}, 1 includes (at, . . . , ak} and 0 excludes {a t, . . . , ak}. By Lemma 30, 9 is indeed an order filter of V. 0 Proof.
32. A finite join-semilattice filter of X is a distributive lattice.
Definition
X is said to be distributive if every principal
Theorem 33. Let X be a finite non-distributive join-semilattice
which is not a lattice, and represent X by means of the mapping a-J, as a set of subsets of its supirreducibles. Every consensus family 9 on X is then an ultrafilter. Proof. The representation
of X as subsets of its set S of sup-irreducibles is not closed under unions or under intersections. By Corollary 14, for any subset A of V, exactly one of A and A’ is in 9. By Corollary 18, 91 contains no disjoint pair of subsets. It follows that 9 is an order filter of V. Since X is not distributive, there is an element a E X such that [a, l] is not distributive. Working in that interval, it is easy to produce elements that satisfy the hypotheses of Theorem 21. It follows that g is an ultrafilter of V. 0
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If X is a distributive finite join-semilattice that is not a lattice, the properties of consensus families on X become much more subtle. First of all, if a 0 element is adjoined to X, then XU (0) becomes a lattice. If that lattice is distributive, then the representation a-J, is closed under unions but not intersections. It follows that there exists a consensus family $9 on X that contains a pair of complementary subsets of V. Further investigation of this will be left for a later paper. On the other hand, if XU (0) is not distributive, then the representation a++J, is not closed under unions or intersections, and much more can be said. Theorem
34. Let X be a finite distributive join-semilattice
that is not a lattice, and represent X as a set of subsets of its sup-irreducibles in the usual manner. If X U { 0) is not distributive, and if X contains at least four atoms, then every consensus family 9 on X is an ultrafilter. Proof. Note first that {Js :gcX}
is not closed under intersection or under union. Hence if 8 is a consensus family, and if A c V, then by Corollary 14, exactly one of A and A’ must be an element of 9. If a consensus family $8 could be found that was not an order filter of V, then there would exist subsets A,B of V such that A c B, A E 97, but Be $21.Hence B’E g, so g contains a pair of disjoint subsets. By hypothesis, X contains distinct atoms a, b,c,d. By Theorem 17, J,U JbC Javb.’ Hence there is an atom c such that clav b, with cc$ {a, b}. By hypothesis, X contains an atom d distinct from {a, b, c}. If d f av b, we have a transitive triple (a, b, c) and an atom d such that (a V b)r\ d = 0. An application of Lemma 30 now contradicts the fact that 9 is not an order filter. Thus we must have dlaV b. But consider the following profile: n(o) = a if a EA; n(a) = b if a E B’; and n(a) = c if (r E B\A. Then both a, and b, are in fa, so a, b E gJR. It follows that d E 93, since dlaV b. But d, = 0, and 0 $9. This final contradiction shows that any consensus family 9J must after all be an order filter. By Corollary 22, any such 91 must be an ultrafilter on v. 0 The reader should note that if X is any finite median semilattice having at least three distinct. atoms a, b, c, then XU (0) is not distributive. For if d is any atom under (avb)A(avc)A(bvc), and dE(a,b,c} we may assume with no loss of generalitythatd=a.Butthend~b~cwithd~{b,c}.Ifd~{a,b,c},weagainhave ds bvc with de {b, c}. In either event, this forces XU (0) to not be distributive. Remark 35. Here is an example that shows that if X has only three atoms, then the conclusion of Theorem 34 need not obtain. Suppose X= {a, b, c, 1 }, where a, b, c are atoms. Let V have three members, and let
~={{1,{2),{3),~1. I Note added in proof: Hence there exists a sup-irreducible SI$ {a, b}. If s>a, we may apply Lemma 30 to the pair (s, a) to deduce that %? is an order filter of K A similar argument applies when s> b. Otherwise, sza
and sz b, so s contains
an atom
c$ {a, b}. The proof
now proceeds
as given in the text.
G.D. Crown et al. / Neutral consensus functions
249
Consider the possible profiles (using symmetry, the following represents every type of profile): n1
=(I, 1,119
7c3 =
(194
a),
712 =(I, 714 =
(La,
776 =
(a,
l,a), 6),
a, b).
Using g as a consensus family, and letting F be the associated consensus method, F(z,)= 1; F(n2)=a since a,=V, b,=c,={1,2}$97; F(ns)=l since ag=V, b,=c,={l}~g; F(n,)=c since a,={1,2}6Q7, b,={1,3}$5Z7, c,={l}~a; F(ns)=l since a,={l}~S?, b,={2}~9, c,={3}~kZI. F(7r6)=b since a,={1,2}BQ!J, b,={3}ES%, c,=O$@. This then produces a consensus family on X that is not an order filter. This leaves open the exact conditions under which Theorem 34 remains true when X has fewer than four atoms.
8. Conclusion We have presented a new model for those neutral consensus methods that are defined in terms of stability families. Included in this model is a notion of duality that plays a key role in understanding the connection between closure under unions and closure under intersections. Our goal was to investigate the properties of stability families that serve to guarantee that every neutral consensus family D be an order filter, be closed under intersections, be a lattice filter, to not have pairs of disjoint sets, and to either contain or fail to contain a pair of complementary subsets of V. This is a project that is still under active investigation. We have demonstrated a clear connection between these ideas, and a generalization of transitivity of relations, and its connection with an abstract version of Arrow’s Theorem. Our results include as special cases much of the standard theory of neutral consensus methods, and thus serve as a natural mathematical model for their study. It has the advantage of presenting the subject in a way that discounts the nature of the objects themselves, but concentrates instead on the manner in which the building blocks in the stability families are put together to form a consensus.
Acknowledgment
The second author’s 90-1008.
research was supported
in part by ONR Grant NOOO14-
G.D. Crown et al. / Neutral consensus functions
250
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