Discrete Mathematics 35 (1981) 173-184
North-Holland Publishing Company
B. MONJARDET
Universit& Paris Vet Centre de Mathimatique
Sociai’e, 54 boukward Raspail, 75270 Paris Cedex
06. France
Received 29 July 1980
We review more or less recent results on metrics associated with valuations or (and)
minimum weighted paths in pose% metric characterizations of posets and a’e point out some
applications of these results to problems invc .!*%nguses of distances in social sciences.
In lattice theory this subject goes back to Glivenko [28-381 and is closely
related to the study of ‘valuations’ or ‘norms [lo, 421. Now, it seems somewhat
obsolete (the recent books on lattice theory do i-lot mention it) and its developments must be found elsewhere, for example in ‘distance geometry’ [12,13], or in
more applied directions [4]. In poset theory, this subject seems to go back to
Haskins and Gudder [32]. But uses of metrices in posets have appeared in several
domains: psychometrics
[27,50], social networks /26], social choice theoiy
[36, 14, IS], datz analysis [2,16-l& 43-451, information theory [20,33,39] etc.
We review results for finite posets along three lines:
(1) Valuations, minimum weighted paths and metrics on posets.
(2) Metric characterizations of posets.
(3) Applications of these results.
(X, S) or X denotes a poset. We denote by !! (0) the greatest (the least)
element of X (if it exists). We denote by [xt ((x];~ the set of all upper bounds
(lower bounds) of x in X. We write xuy = [x) n [y); xly = (x] n (y].
For x and y in X, we write x<y if y covers x (x<y and x<ySz
implies
y = z). The couering graph. 3(x)
of X is the (simple undirected) graph whcse
vertices are the elements of X and whose edges ae these pairs (x, y) satisfying
X-C y or y < x ; such an edge .:: also denoted by a. Fior x, y in X, a path from x to y
is a path frcm x to y in Gc:C), i.e. a sequence e, : x =x0, xi,. . ,., Xi, . . . , X, = y.
such that (x,, x1)7 . . . , (x, _I’l x,) are edges in d;(X): the length of such a path is p.
X is connected iff for all x, y in K there exists a qath from x to y-
B. Monjardet
174
I,et X be a set and il a non-negative real function defined on the set J? of
ordered pair!, of X. We define properties which can be satisfied by d:
(1) For ev’ery XEX, d(x,x)=O.
(2) For all x, y E X, d(x, y) = 0 implies x = y.
(3) For all x, y E X, d(x, y) = d(y, x).
(4) For all x, y, z EX, d(x, y)ad(x, t)+d(z, 9~).
(5) For aI1 xl,. . . , xk, yl, . . . !,Yk+lE X,
‘z
d(xi, xi) + ,_,C2k+Id(yi,
Irzi<jSk
--.
Yi)d
.--
C
1siSk
C
l6j<k+l
d(q, Yi)*
(6) For aA x, y, z, TV X, d(x, y)+d(z, t)~max(d(x, t)+d(y, t), dlx, t)+d(y, z)].
(7) For all x, y, z EX, d(x, y)~max[d(x,
z), d(z, y)J.
The function d is a semimetric, quasimetric, metric, k-hypemretric, tree-mefric,
ultra,nctric iff d satisfies the properties [(l), (2), (3)], [(l), (3), (4)], [(l), (2), (3).
(411, UU, (‘9, (31, (5% [W, (21, (3), WI, Rl), G3, (31, (711,respectively.d is an
hypermetric iff it is k-hypermetric
for all positive integers. Note that a lhypermetric function is a metric. Hypermetrics have been defined by Deza [S3]
about the symmetric difference distance between sets. Tree-metrics (or metrics
satic,fying the four point property) have been introduced by Eiuneman [21]. The
following irreversible implications are shown by Kelly [34,35]:
d ultrametric + d tree-metric
3 d hypermetric + d metric.
In the following sections, X is always a finite (and connected) poset. We just
plint out t’hat many results are valid ‘or more general posets, such as locally finite
pose& 0; (and) directed posets.
3. Vre’luaths ad
metics on a poset
Let .Y be a poset with 1 and u a real map defined on X. For x and Y in X, we
write
c ‘.(x, y) = z,Tizn_Yv(2),
(xuy),. = {z E xuy with u(z) = v+(x, ~11,
-. -
The map t) is called an upper valuation it?
for all x, y, .z in X with z <xx, zcy,
u~x!+u(y>~v+:x,y~+u~z~,
Note t’hat an upper valuation is not necessarily isotone. Now, we will always
assum- t’ isotone, or, equivalently, C-I,,
20. Then, WC have:
For ar; isotoPte map 11 defined
0,~ X, the following statements are
equivaleizt :
t
1) 21 is nn upper ualuatiorz.
!21 For ~20 x, y. z, in X, u’ix, 2)-t- u+(z. y)a 0+(x, y)+v(z).
(3~ For all x, y, z, in X, d,.fx, y)~d~,fx, z!+&(z, y).
175
Metrics on partidly ordered sets
For an edge u = (x, y) in the covering graph G(X) of X we define the weight of
u by w(u) = Iv(x)- u(y)]. For x, y in X and p,.., a path from x to y, we define the
weight of Px,, by w(P,,) =cUSp, w(u) and we write
&(x, y) = minimum
weight of a path from x; to y.
Clearly, $ is a quasimetric on X.
quasimetric (minimum weighted path).
X, one obtains the classical shortest
w(P,,) = 8,(x, y) is called L v-geodesic
For a (isotone) real map L‘defined
&(x, y&&(x,
y):
xsy
We call such a quasimetric, a M.W.P.
If u is the rank function of a graded poset
path metric on G(X). A path such that
from x to y.
on X, we have clearly
implies
&(x, y)=d,(x.
y)=u(y)--U(X).
Tht?orem 2. Let X be cr poser with 1 n:J u an isotone (a stictly isotone) real map
&fined on X. The following statements are equivalent:
(1) v is an up&pervaluation.
(2) d, is a quasimetric (a. metic).
(3) 4, =z8”.
(4) For all x, y there exists z E (xwy), such that $(x, y) = if&(x,z)+&(z, y)_
Statement (4) means :!rer e exists a v-geodesic from x to y ‘through (xuy),.‘. It
can also be replaced by- for all x, y, z with z E (xuy),, 8,(x, y) = 8,(x, z) -t.&(z, y).
Theorem 2 [5] generalizes previous results obtained in the case where X is a
semi-modular poset [32], a jo+ semilattice [54, 241, a lattice [16, 18, 311 or a
ljormed lattice [28].
For example, in the case where X is a join-semilattice the statements (1) and
(4) can be written:
(1) For all x, Y, L with ZSX, z~y, ~~(x)+v(y)~v(xvy)+v(z).
(2) For 211x, y, &(x, y)=&,(x,xvy)+S,(xvy,
y).
Moreover, these stamments are equivalent to:
(5) For all x, y, z, v(~~~)fa)(~~y)~v(xvy)+~~(~).
(6) For all x, y, z, 4(x, y)a$(xvz,
yvz).
(7) For all x, y, z with y =Zx, 4(.x, y)~ 4,(x v z, y v z).
(8) For all x, y, z, t, d,, (x, y) + d, (z, t) 3 d,(x v z, y v 1).
The associated quasimetric is 4(x, y) = 2v(x v y) - v(x) - u(y). In the case
where X is a ‘lattice (and ZI isstone), the statement (1) is equivalent to:
(1’) For all x, y, v(x)+v(yjav(x\~yj+v(xr\y)
and, thus, an isotone upper valuation is nothing else that an isotone sernimodular
((Jr submodular) map.
Let us point out the following variations of Theorem 2:
map v satisfying v(x)+ v(y)S~+(x, y)+ v(z) for every z i
ently, v(x)+ v(y)-2u+(x, y) is a quasimetric); second with
that U(X)-I-a,!),) s
isotone ‘lower v~~~~~~~~’ i.e. an (isotone) map u su
equivalent
v-(x, y)+ v(t) f*i every z in xsay (then,
y, v(x:l+n(y)--2v--(x, Y) is a
R . Monjardet
176
quasimetric; here, u-(x, y) = max, Sx,ZSy~(2)). If X is a poset witli 0 and 1, we call
(isotone) valuation an (isotone) upper aud lower valuation. Then, equivalently,
or
u(x)+~~(y)=u+(x,y)+v
(x,y),
or u +(x, y ) - u-(x, y ) is a quasimetric,
6,(x, y) = V+(x, y)- V(J”, y).
In the case where X is a semilattice, the aboki; ?1;,-simebnc:j can be ust:d 10
define ‘normalized’ quasimetrics. An upper valuation is said to be positive i,ff, for
every x in .X, u(x) 20 and u(x) = 0 iff x is a minimal element in X. If u is an
isotone positive upper valuation and d, the associated quasimetric, we write
dcl(x,Y)’
4(x, Yb
O(xVy) if
i 0
u(xvyj>(),
if u(x v y) = 0.
~~gosition 3. Let X be a join-semilattice and u an isotone pmitiue upper valuation
on X. Then, d; is a quasimetric with for all x, y, d;(x, y)~2,
and d$x, y) = 1 ifi
u(x v y) = u(x)+ u(y). Moreouer, if X is u lattice, d:,(x, y)C 1. for all x, y.
This restilt L-193generalizes constructions of normalized metrics on a boolean
lattice 127,401, or a partition lattice [49, 371 (see also [17]). Note that in a lattice
d:(x, y) = 1 implies x A y = 0 with the converse implication if u is a (positive)
valuation.
There exists other means to define metrics from valuations. For example, if u is
an (isotone) strictly positive (u(x)>O, for every x) upper vzlluation on a poset X
with 1, Log t: is an upper valuation, and, thus Log[(u’(x, Y))~/_(x) u(y)] is a
quasimerric on X. (Set [6] for many such results).
The next result allows to characterize the metrics defined in Theorem 2. Let X
be a posst with 1, and d, associated with an isotone real map u clefined on X.
Clearly, d,. satisfies the two following botweenness properties:
l
13,: for all x,y,z
with xs~-~y,
B,: for all x, y, z with z E (xuy),.,
d,(x,y)=&(x,
~)+&,(z,y),
d,(x, y) = d,(x, t)+ d,(y, z).
Let X be a poset with 1 and (2 a real map defined 011 X’.
We wriF4
c(x) = -d(x, l)+Cte. I_fd satisfies the properties B, and B,, then d = d,. Moreover, d
is rm:-negative, a quasimetric, a metric ifl u is isotone, an isotone upper valuatioat, a
5;, k Ly isotorre upper valuation, respectively.
Thus. the metrics d associated with an upper valtlation are characterized by the
two betweenness properties
and B,. Note that the upper valuation u associated
to d is unique up to constant. Also, it is sufficient to assume B, with y = 1.
is a join+emiiattice,
B, can be replaced by
Metricson partially ordered sets
177
. Let X be a join-semilattice and d a rea! map defined on Js?. There
exists an isotone real map v defined on X such that d = 11, ifl d is non-negative ai;d
satisfiesthe propertiesB, and B,. Moreover, d is a quasimetric (a metric) i,jj’vis an
upper valuation (a strictly isotone upper valuation).
Note that, in this corollary, the g:>perty “v is an upper valuation”‘ can 13e
expressed by the properties (5), (6), (7) or (8j (equnivalent to the statements (l)-(4)
of Theorem 2) of the map d (=d,,).
For a poset with 0, or a meet-semilattice, we need the dual properties:
B*: for all x, y, z with z E (xly),,
R,: for all x, y,
4(x, y) = h(z, X) + &(z, y),
$u(x, y) = d (x A y, x) + JJx A y, y).
For a poset with 0 and 1, we obtain an interesting
improvement:
Cot&My 6. Let ,X be a poset with 0 and 1, and d a non-negative real map defined
on XT We write v(x)= -d(x, l)+Cte (or v(x)=d(O, x)+Cte). If d satisfies the
three betweenness properties B,, B, and B,, then v is an isotone valuation and
d = 4 is tile associated quasimetric.
Proposition 4 and its corollaries improve results of Rarthilemy
and Smiley [54].
[j] arid Wilcox
A poset X is called graded iff there exists an integer-valued function r defined
on X, such that x < y and r(y) = r(x)+ 1 iff x < y. Thus, r is strictly isotone.
Note that for a poset X graded by r, S, delined as in the Section 3 is nc-{thing
else that the shortest path distance in G(X); we will denote by ti this distance.
The following assertion is clear.
A poset X with 1 is graded if and only if there exists a metric d on
satisfying B, *and d (x, y) = 1 if x 4 ye
A poset X is called c:pper seminlodular i
SW
x $ y, there exists t E
semimodular poset. A poset is called ~lod~l~r iff it is
se
odular .
B. lbhjardet
178
;*quivalent:
(1) X is upper semimoduiar.
(2) I’ is an upper valuation.
(3) d is a metric.
(3) d, = s.
(5) For all x, y, z with 2 E (xuy),, 6(x, y) = S(x, 2) + S(z, y).
(6) There exikts a metric d on X satisfying &,
d(x,y)=l
ifxty
Moreover, in (6), d = S.
B, HOPu(x) = -d(x, 1) + C te) and
Note it is not necessary in (6) to assume X graded.
For a poset X with 0 and 2, graded by r, the following statements are
equivalent:
(1) X is modular.
(2) r is a valuation.
(3) For all x, y, r(x) + rl V) = r+(x, y) + r-(x, y).
(4) d, and [r(x) + r(y) -- 2r-(x, y)] are metrics.
(5) For all x, y, z, t with z E (xuy), t!znd t E (xly),,
6(x, y) = 6(x, z) + S(Z, y) I=
ax, t)+ m,y)*
(6) There exi.sts a non-negatiue real map d on X satisfying &,
u(x) -5 d(0, x)) and d(x, y) = 1 if xx y.
Moreover, in (h), d = 6.
B,, I$ CfcH
An example of modular (non-latticial) poset is given by the ‘dominance
il] in the set of number-p;Mions
of an integer [S].
order’
rprgr3‘0. Let X be a join-semilattice. The following statements are equivalent:
( 1) X is upper semimodular.
(2) X is graded and for all x, y 6(x, y) = 6(x, x’v y)+ S(x v y, y).
(3) There exists a metic d on X satisfying B<, B, and d(x, y) = 1 if X-C.y.
Moreover, in (3), d equr.ls 6.
Theorem 8 [46] generalizes results of Corollary 9 [32], these two references
containing also o*t:her characterizations of semimodular posets. The more or less
classical characterizations of (upper) semimodular lattices or modular lattices by
properties of r or 8 are particular cases of these results. The equivalence (9) and
(3) in Corollary 10 [5] is a clear consequence of Corollary 5.
In a (upper) semimodular poset, there exists an interesting criterion to recognize an (upperj valuation. Let v be an isotone real map on a poset X with 1. The
map v satisfies the. quadrilateral condition iff:
for all x, y, 2,t in X, with t<xc, t<y,
U(X)-t U(Y’)2 u(t) -I-U(Z)
3.
xdz,
y<z,x#y,
Metrics on partially ordered sets
179
t v be an isotone real map on an upper semimodular poset (an
upper semimodular join-semilattice); v is an upper ?jaluation if (if and only if) v
satisfies the quadrilateral condition.
We shall now turn to characterizations of modular and distributive lattices bY
properties of valuations or metrics. In a lattice L a valuation is defined bY
v(x)+v(y) = v(xvy)+ v(xny), and if v is a real map on L, we define dU(x, y) bY
v(x v y) - v(x A y).
(1)
(2)
(3)
(4)
Let L be a lattice. 7’he following statements are equivalent:
L is modular.
There exists a strictly isotone valuation defined on L.
There exists a strictlyisolone real map v on L suck that d&x, y) is a metric.
There exists a strictly isotone real map v on L such that for all!x, y, z in H-?
dJx, Y)~~(xvz,Yvt)+d,(X:AZ,YAt).
(5) There exists a strictlyisotone real mnp v on L such that for all x, y, z, t in L,
dub,y)+&(z, t?ad, (xvz, yvt)+d,(x/\z,
yr\t).
(6) There exists a nonnegative real map d on L2 satisfjing B,, B,, 63, and
dk y)#O if x#y*
(7) 7%ere exists a metric d on L satisfying B<, B,, B, and d(x, y) = 1 if x 4 y.
The characterizations (2) and (3) of modular lattices are well known (see [ll]
and lead easily to (4) and (5). The metric characterizations (6) and (7) are obvious
consequenG,_
--Q of <orollary 6; in (51, d = 6. Note that the properties Be, B, and
B, are implied by the following interval betweenness property (with d(x, y) =:
d(y, x) for x < y):
B: for all x, y, ZEX, z ~[x~y,xvy]
implies d(x,y)=d(x,z)+d(z,y).
In a lattice L a valuation is said to be distributive if it satisfies the following
equivalent properties:
(a) For all x, y, 2 in L,
v(xvyvz)=v(x~+
(b 1 For all x, y, z in L,
B. Morijardet
180
(4) There exists a strictly isotone zaluation 21on L such that for all x, y, z, t in L
t&(x, y)+&(z,
t)Gdu(xv:!,
y/\t)+d,(x/\z,
yvt).
(5) There exists a strictly isotone ml~aalion on L such that 4, is 2-hypemet&
(or
hypvmetric).
(6) There exists a nonnegative map d defined on L* satisfyin.: B, x -Cy implies
d(x, y) = d(y, x) and d(x, y) ok0 if xf y.
(7) There exists a metric d defined on L satisfying EJ and d(x, y) = 1 if x< y.
(8) T&re exists; a nonnegative nznp d defined on L* satisfying B and d (x, y ) f 0
The t:haracterizations of distributive lattices bjr (2) or (3) are classical (see, for
example, [ll, 41. The characterizations (4) and (5) are due to McDiannid [41] and
Kelly [34], respectively. (6) and (7) are easily obtained from (3) and the characterization (6) of modular lattices (Theorem 12). In (6) d is a distance (associated
with an arbitrary valuation); in (7) d = S. The characterization (8) [3] does not
imply that d is symmetrical: here $(d(x, y ) + d(y, x)) is a distance (associated with
a valuation).
s 14. (1) All possible valuations on a modular lattice are detelmined (see
[ :kI]. The strictly isotone positive valuations on a distributive l,,tice ‘T‘eorresponds biunivocally with the positive measures nl on the set J of join-irreducib!es
(+CH of 7’: U(X)= r~((y E J’: y s x}). Thus, the associated metrics are metrics
defined by the measure of the symmetric difference in a ring of sets (and thus are
hyperme tries).
(2) ‘There exists characterizations of the metric spaces defined by modular or
distributive lattices [28,29] related with the betweenness characterizations
of
these lattices (see [30, 11,4,13]j.
(3) The classical characterization of distributive lattices by the existence of a
ternary
median
(see [ll])
can be broadly
extended.
Moreover,
if
(X1,. . . , Xi,. . . , X,) is a n-t@
iif elements of the distributive lattice T, the
generalized medians of (x,, . . . , q, . . . , x,,) are the elements in 2’ minimizing the
quantity I:_, d(x, q ). G-here d is a distance associated with any valuation on D
[3,48].
(4) The above results allows to characterize shortest path metrics 8 in
segimodulsr posets. The characterization problem of such metrics in other graded
or arbitrary posets see J-ISopen. Let us point out the l-elated works on the covering
elding result [9]; a
graphs of iattices or posets (see especially [2,C]) and the
connected pcset is determined to within isomorphism by 8 and its ‘fence metric.
We end this asection with tzt? related considerations. A tree is a (non-directe
connected graph without cycles. XC metric characterization of flees by Buneman
c iollowing result.
Metrics on purtiafIy on&red sets
. For a connected pose? X, the covering graph
only if the (shortest path) metric 6 on X is a tree - metric.
181
G(X) is a tree if and
A posei X is said a tree pose? iff X has a 1 and (1, G(X) is a tree or (2) every
element x in X, x # 1 is covered by an unique element or (3) every interval of X is
a chain. Thus a tree poset is an upper semimodular join-semilattice.
A real isotone map u defined on a poset with 1 is an ultrametric upper valuation
iff: for all x, y, 2 in X, u+(x, y) Smax [0+(x, z), v-+(2, y)].
Clearly, v is an upper valuation and for every real number c, the restriction of
the associated dis;ance d, to X, = {x in X such that u(x) = c} is a quasiultrametric.
position 16. Let X be a poset with 1. The followirzg statements are equivalent:
(1) X is a tree pose?.
(2) X is a join-semilattice with for all x, y, z 1(xv y, x v z, y v z}\ s 2
(3) Every isotone real map defined on X is an ultra~etric upper valuation.
(4) Every isotone real map defined on X is an upper t&u&on.
(5) There exists an isotone ultralmetric upper valuation defined on X.
(6) X is graded by an ultrametric upper valuation.
(7) Tke (shortest path) distance S on X is a tree-metric.
The characterization (2) is due to Leclerc; the characterizations (3), (4), (S), (6)
are due to Barthklemy [5, 61; the characterization (7) is an obvious consequence
of Proposition 15. As by product one generalizes a well-known construction of
ultrametrics in math,smatical taxonomy (X0 equals the set of minimal elements of
X, and for x, y in X(,, d&, y) = r (xv y)).
5. Appkations
Social sciences use more and more ‘discrete’ structures to describ*z data or
models. For example, social or individual preferences, taxonomies, social networks, kinship systems, linguistic models, games, decisionsi etc. tan be formalized
by slph structures, at least partially. The problem to compare these data or these
models occurs naturally; for example, one need to compare resuljts :“rom different
paired comparisons experiments. A natural idea to compare discrete structures is
to compur:e a. distance based on the minimum number of admissible transformations needed to transform a str cture into anot
discrete structures in a particul
minimum weighted path metric;
poser (see especial1
generalized isformatior. theory) is ciosc!,11 related to the study of antitone real
maps on a poset [33,39]. T!lese facts explain why ihc results of Section 3 can be
applied. The :ase of relational str:lctures is a good example, because many sets of
relations used in social sciences are lat.tices or semilattices,
often upper
semimodular (complete relations, complete preorders, equivalences) or lower
semimoduls. (antisymmetric relations, partial orders, weak orders). So cla~sicd
distances (fo examp!e, the symmetric difference distance) and many other distances between relations can be obtained from the above results ([16,17] for
partitions; [6,7] especially for p-reorders or complete preorders). For applications
to mathema ical taxonomy, we shall quote the Boorman-Oliver
paper on metrics
between finite trees [lF(] and the Leclerc result on the semimodularity
of the
lattice of ultrametrics [38, 561.
Ali<:t!?er &ssical problem occurrilng in social sciences is to summarize or to fit at
best numerous data. A classical approach is to search the summary or the fitting at
‘minimum distance’ from the data. F;or example, the classical statistical means or
medians are obtained in this manner. The theory oE the median procedure for
binary relations, closely related to the theory of generalized medians in a
distributive lattice, is a framework subsuming many approaches to these problems
for relational data (see [S]); note that it leads to difficult combinatorial
or
optimization problems. Another interesting example is the use of the lattice
distance in the join-semilattice of complete preorders to fit an ultradissimilarity or
a partition to a dissimiiarity relation (i.e. a complete preorder on X2: [51,52]).
A natural question occurs whlzn one use distances in order to compare, to
summarize or to fit data: wh,ich distance should be chosen? A (partial) answer is
given by the systematic study of properties of metrics, in particular their axiomatic
characterizations. The first result seems go back to Kemeny [36] for the symmetric difference dist.lnce between complete preorders. Other characterizatit>ns were
obtained by Mirkin and Cherny 145,431 for equivalences and by Bogart [14] for
partial orders and asymmetric relations [15]. The results of Section 4 allows
significant improvements and extensions of these characterizations
[6,7]. Note
especially that the triangular i;lequality property which is not necessarily a
relevant condition in social SC~C.IW,is very often implied by other properties.
[ I! X!. Aigncr. Combinatorial
Theory (Springer-Verlag,
New York, 1979).
121 P. Arahie and 5.4. Boorman,
Multidimensional
scaling of measures
of distance
between
partitions. J. Math. Physchol. 10 (1973) 138-203.
[31 M. Barhur. M&i.‘ane, distributivite,
eloigncments.
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