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On permutation lattices
Mathematical Social Sciences 27 (1994) 73-89 0165-4896/94/$07.00
fQ 1994 - Elsevier
73
Science
B.V. All rights
reserved
On permutation lattices Vincent Duquenne” URA, CNRS 1294, G&attique, Neurogt%tique 75006 Paris, France
Ameziane
et Comportement,
45 Rue des Saints P&es,
Cherfouh
VFR de Mathtmatiques,
Vniversitt de Paris V, 12 Rue Cujas, 75005 Paris, France
Communicated by M.F. Janowitz Received 3 February 1992 Revised
14 August
1993
Abstract The lattice perm(n) of permutations of an n-element set n = (1,. . , n}, ‘rooted’ at (1,. , n), is shown to be meet- and join-semidistributive, which implies known results such as the non-existence of M,-sublattices, and that the complementation defines a congruence relation in Perm(n) with 2”-’ classes. The meet-core of Perm(n) is shown to be the set of meet-irreducible elements together with all the elements that have two upper covers that moreover generate a covering sublattice isomorphic to Perm(3); this shows that the meet operation is completely expressible in terms of reversing adjacent pairs. A recursive construction of Perm(n) as a Galois lattice -via a kind of summation process - is given, which has been a key for obtaining clear drawings of Perm(4) and Perm(5) with our new PC/VGA graphic program GLAD (General Lattice Analysis and Design). Finally, the congruence lattice C(Perm(n)) is recursively characterized through the order of its meet-irreducible elements. Key words:
Permutation;
of collapsing;
Lattices
semidistributivity;
Meet/ join core;
Galois
lattice
representation;
of congruences
List of symbols n = (1,. , . ) n}
a finite set permutations I-compatible pairs cardinal of. , . cover, order relations set operations lattice operations
a, P, Y . . f p,(a) I I -<,
<, i
C, 5 u, n, u, v, A, v, A * Corresponding
SSDI
n
author
0165-4896(93)00733-B
at: CNRS,
10-b rue A. Payen,
F75015
Paris,
France.
Heredity
V. Duquenne,
74
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
E, CL)
meet-essential elements meet-core a decreasing mapping subsets of C(Perm(n)) disjoint union (weak) perspectivities congruence
K,(L) A b
A’ = { y E Y ( aRy all a E A}
“H(n), Y&, ?i&. . . fi t ’ LT 5,
@(my m)
0. Introduction Permutations on a set play a prominent role in many fields of mathematics, as well as more applied topics like statistics, data analysis, up to experimental designs and genetics, or kinship systems in anthropology. The first property which is emphasized in every book is that starting from any permutation I = (123. . .) on a set n={1,2,..., n}, any permutation can be obtained by a series of transpositions reversing neighbors (see Fig. 1). This gives rise to the symmetric group S, on an n element set. It was observed long ago in the context of Social Choice (See Guilbaud and Rosenstiehl, 1963, and Barbut and Monjardet, 1970) that the transpositions also define the cover relation of a lattice, which is denoted by Perm(n). If the group-theoretic properties of permutations are well known, some lattice properties of Perm(n) are not so clarified yet. This paper aims at contributing to fill the gap and to the revival of interest in permutation lattices (Chaumeni Nembua, 1989; Le Conte de Poly-Barbut, 1990; Markowsky, 1991; Bennett and Birkhoff, 1992).
1. Lattice properties
of Perm(n)
Let n = {1,2, . . . , n}, Perm(n) be the set of all permutations on n, I = (123. . .) and T = (. . .321) be elements of Perm(n). For (YE Perm(n), let p,(n) 321
231 /ok0
21:\/
312
I
I 0
’ 12
12
132
23
I$ Fig.
1. Perm(3)
is ‘rooted’
at 123. The edges
are labelled
by the neighboring
locations.
V. Duquenne,
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I Mathematical Social Sciences 27 (1994)
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73-89
be the set of ordered pairs of distinct elements that are I-compatible in (Y,i.e. in the same order as in I. T-compatible ordered pairs and pT(o) are defined dually. Two ordered pairs of neighbors (i.e. elements at neighboring locations in the permutation) are adjacent if they share some element. It is often worthwhile viewing a permutation simply as a succession of maximal blocks of either I- or T-compatible adjacent pairs. p,(o) and pT(cx) are subsets of p,(l) and pT(T), respectively. Since any ordered pair (x, y) E n x n is either I-compatible or T-compatible, the duality between pT(o) and p,(o) may be stated as follows: PET
(*)
and
= PT(T)WPI(~))
p,(o)
= pI(WJWp,(4)
7
where Tr(R) stands for the transpose of a relation R. When p is obtained by reversing a I-compatible pair of neighbors in (Y, we write LY--< /3: let 5 be the transitive closure of the relation ‘--< or =‘. Thus, (Perm(n), 5) is the order of all permutations of n rooted at I, and is ranked since : foranycu c/3, ff --
(**>
iff /p,(o)]
= Ip,(p)]
-1.
Generalizations of this construction are studied at length in Bennett and Birkhoff (1992), while algorithmic aspects of Perm(n) are developed in Markowsky (1991). The set M c Perm(n) of permutations that have only one upper cover is characterized as follows: each p E M has exactly one I-compatible pair (x, y) of neighbors. The set J of permutations having only one lower cover is characterized dually. A lattice is meet-semidistributive (SD,,) if x A a = x A b 3 x A a = x A (a v b). Join-semidistributive lattices (SD,) are defined dually. R* is the transitive closure of a relation R. For standard notions on lattices, see Birkhoff (1967), Barbut and Monjardet (1970), or Crawley and Dilworth (1973). Proofs that (Perm(n), 5) is a lattice have been given in Guilbaud and Rosenstiehl (1963), Yanagimoto and Okamoto (1969), and in the more algorithmic construction given recently in Markowsky (1991): Theorem p,(o)
1. (Perm(n),
uPI(P
5) is a ranked compEemented lattice such that pI(o
and ~,(a
v P) = (~~(4
UPAP))*
A p)
=
hold.
Lemma 2. Let cy E Perm(n) and (Y+ = v { a0 E Perm(n) ( a0 >- a}. Then [a, (Y‘1 z X iEl Perm( IDi\) holds for the family (Bi);,, of maximal blocks made of I compatible adjacent pairs of (Y. Proof. Let cxi be obtained by reversing the blocks Bi of cy, for all i E Z. [a, ai] s Perm((B,[) holds, obviously. Let p E [ a, CY+] and pi= /3 A cxi (iEZ), so that p =‘ V iE, pi holds. Each pi is obtained from (Yby reversing the pairs of Bi that are also contained in pl(/3). Since Perm(n) is ranked, this implies in turn that
/? = V iEI pi holds, which shows that the mapping /3 * ( pi)igl is bijective; since it is, moreover, a 01-meet-preserving map [a, a+]--, Xi,,Perm(\B,(), this is a q lattice isomorphism, so that [(Y, (~‘1 z xiEI[ a, ai].
76
K Duquenne, A. Cherfouh I MathematicalSocial Sciences 27 (1994) 73-89
3241 3214
\I:
,‘-g:
1: 3,243
f/-l324
/
-c:
2314
2134
12w
Fig. 2. The permutation lattice Perm(l) ‘rooted’ at 1234 (V and A indicate join- and meet-irreducible elements, respectively).
In particular, Lemma 2 generalizes known properties (see Chaumeni Nembua, or [cx, cq v a2) E 1989): for LX-
3. (Perm(n),
5) is a meet- and join-semidistributive
lattice.
Proof.
Let x, (Y,p E Perm(n) be such that x A (Y= x A j? = x A cy A j3 ; in order to establish SD,, , it must be shown that: (*) x A cx = ,y A (a v /!I) holds. Claim 1. (*) holds when (YA p -< (Y, p. Let y be such that x A (Y --< y 5 x, and let CY,p be obtained from cx A j? by reversing (a, x) and (y, b), so that a < x 5 y < b holds. By Lemma 2, [LYA /3, (Y v /3] is isomorphic either to C, X C, when x # y, or to Perm(3) otherwise when x = y. Suppose that [(YA p, cy v ,6] = C, x C, holds; (Yv /3 is obtained from LYA p by reversing (a, X) as well as (Y, 6); hence P,(Y A (a v P)) = (P,(Y) U ~,(a v P))* = (P,(Y) U P~(” A P) ’ ((6 x)3 (Y, b)))* = ((P,(Y) U p,(a)) n (P,(Y) UP,(P)))* = (PI(Y A a) n P,(Y A P))* = P~(Y A a), since I~,(r)l = I(P~(Y A a)I - 1 impliesp,(y * Q’) =P,(Y) UP,(~) andp,(y A P) = P,(Y) Up,(P). Otherwise, when [a A P, (Yv P] z Perm(S), LYv p is obtained from LYA p by reversing the block (a X 6);
thus, P,(Y x), (x, b),
* (a ” P)) = (P,(Y) u PL(a ” P))* = (P,(Y) (a, b)})*; now, y >- y A ct’, y$cx, (Y transposing
(x, a)@pL(y), by transitivity, P>>* = /‘I(?’
so
that (a, x) Ep,(y); similarly, (x, b) up,, Hence, pI(y A (cy v p)) =
(a, b)Ep.(y).
A a A p).
u P,((y (a, x) in
A P) \ {(a,
(YA /3 imply which implies,
(p,(y)
U P~(‘Y A
V. Duquenne,
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I Mathematical Social Sciences 27 (1994) 73-89
77
Claim 2. For x, (Y, /3 such that x A (Y= X A p there exists some y >- (Ywith y 5 LYv p and x A y = x A (YA p. Let (Y, and PI be such that (YA p --< CY,9 (Y and (YA p -< & i p and S, = [a A p, czl v PI]; then: either S, z C, X C, or S, z I-I6hold, and (*l) x A (Vs,) = x A (Y A j3 holds, by Claim 1; if vS, 2 (Y,then Claim 2 holds; otherwise, Q $ vS,, cxl 5 a and czl < Vs, implies (YA (VS,)# AS,; let CQ, & >- (YA (VS,) be such that (Y*5 (Y and &E S,. Let S, = [a A (VS,), cx2v p,]; (*l) and Claim 1 imply that similarly: (*2) x A (VS,) = ,y A cy A p holds. By reiterating this process taking each (Y~along a maximal chain of [a A /3, a], Claim 2 is reached by taking y >- a! in the first Si with V Si > (Y. By induction on the rank of y along a maximal chain of [(Y, (Yv p], the SD h law is proved. The SD, law is established dually. Cl
Many properties of Perm(n) are consequences of the SD laws, such is the absence of M,-sublattices (with 5 elements, 3 atoms). On the other hand, as observed for Perm(n) by Le Conte de Poly-Barbut (1990) and Markowsky (1991) (see below, Fig. 8), and more generally for pseudo-complemented lattices by Chaumeni Nembua (1989)) Chaumeni Nembua and Monjardet (1992) one obtains: Proposition 4. Let L be a complemented SD, and SD, lattice, A and C be its sets of atoms and coatoms, and let 0 be the relation ‘sharing a complement’. 0 is a lattice congruence relation on L having 2 IAt= 2”’ classes, of which the minimal
(maximal) elements are the joins of atoms (the meet of coatoms, respectively).
2. The core of the permutation
lattice
The core of (arbitrary) finite lattices has been introduced in Duquenne (1987) for dealing with the following question: What minimum piece of information must be added to the order of meet-irreducible elements (M(L), 5) of a lattice L in order to be in a position to reconstruct it? For traditional classes of lattices (e.g. geometric, modular, join/meet semidistributive) the meet- and join-cores have been characterized (Duquenne, 1991), by strengthening lemmas that depend on the semimodular laws. In the case of the permutation lattice, as opposed to these classes, since semimodular laws are not available a tailor-made approach has to be developed. Let us first recall some definitions (see Duquenne, 1987, 1991): an element x E L\M(L) is said to be meet-essential when the following holds: (ME).
there exists some proper order filter X of [x)\(x) for which AX = x and such that for any Y G X, either A Y = x or A YE X holds
We say in short that the triple (x, X, a) is meet-essential when the condition ME holds for (x, X) and a >- x ‘is not captured’: a E [x&Y. Hence, x is meetessential iff such a triple (x, X, a) exists. For any y E L\(x], let yzx = A {z E x 1
V. Duquenne,
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A. Cherfouh
I Mathematical Social Sciences 27 (1994)
73-89
z 2 y} be the (unique) smallest element of X greater than y. The set of meet-essential elements is denoted by E,(L) and the meet-core is defined by: K,(L) = M(L) U E,(L). Theorem [Duquenne, 19871. For a finite lattice L and PC L, the mapping z-+ [z) fl P (z E L) is an isomorphism L+ .9(P) onto the filter lattice of the partial meet-semilattice defined by P in L, iff P _> K,(L).
The following lemma is needed to study arbitrary cores: N,-M, Lemma [Duquenne, 19911. Let (x, X, a) be a meet-essential triple in L, and let x,, x,EminX for which x,#x,‘.a’X= r\{yEXlyra}. Then, {x,, x2, a, a’“} generates one of the sublattices shown in Fig. 3. Lemma
5. Let (x, X, a) be a meet-essential triple of SD, lattice. Then, a’x E {x, a, azX, x,, a v x1) z N5
min X holds, and moreover for any x1 E min x\{a’“}, holds.
Proof. In cases (i), (ii), (iv), (v) and (vi) of the N,-M, Lemma, the inequality x2 A x1 = x2 A a # x2 A (x1 v a) = x2 holds, contradicting the SD h law. Hence only case (iii) can occur in Perm(n). 0 Lemma 6. Zf an element x E L of an arbitrary lattice L has only two upper covers, x1, x2 >- x, which moreover belong to a sublattice H6 g Perm(3), then x is a meet-essential element in L.
Let y,, y, be such that x, < y, and x2 < y2 in this H6 sublattice. Let an d zl, z2 E X. We claim that z1 A z2 E X or z1 A z2 = x hold: if X=[Y~)~[Y,) zr, z2 2 y, or zl, z2 2 y,, this is trivial. Suppose therefore that z1 E [yI)\[y2) and (Z, A z2 P x, or z1 A ~2 E [~2)\[~1) hold; ( z1 A z2 jZ’X and z1 A z2 #x) iI@ieS z2 2 x1), since x has only these two upper covers in L; suppose that z1 A z2 2 x1 holds; then x2 < z2 and x1 % z1 A z2 < z2 imply x1 v x2 = y1 v y, 5 z2, contradictProof.
ing z2 E [Y&Y&
0
Theorem 7. A permutation x is meet-essential in Perm(n) a, b >- x, such that [x, a v b] is isomorphic to Perm(3).
iff it has two covers,
The sufficiency is implied by Lemma 6. Conversely, let (x, X, a) be a meet-essential triple. By Theorem 3 and Lemma 5 x -< a < a* E min X, and there exists at least one x1 $a’” in min X such that N = {x, x1, a, aIX, x1 v a} is a N5 sublattice, since (min Xl 2 2. Claim 1. There exists b >- x for which [x, b v a] G H6 and is covering in Perm(n) If x -< xi, take b = x1; this is true by Lemma 2. Otherwise, let b >- x be such that b < x1 ; the triple (x, X, b) is meet-essential and bIX = x, holds, so Proof.
V. Duquenne,
x,va=x
A, Cherfouh
va
I Mathematical Social Sciences 27 (1994)
:X
x
79
73-89
va
a:X
2X
f< *1/‘\./\, 8
.I/+=\
.,:
\
x2
..’ /.” .‘I
1 # x2
a
m *s ...I (2
,J
I cf.‘.. X
\
,,..,O ..’‘.
d” X
(ii)
Gil
Fig. 3.
-< b
80
V. Duquenne, A. Cherfouh I Mathematical Social Sciences 27 (1994) 73-89
contrary would also imply that two distinct covering H, sublattices element chain, which is impossible. q
share a 3
Corollary 8. A permutation is meet-essential in Perm(n) iff it has exactly two pairs of neighbors that are adjacent and I-compatible.
Hence, all the equalities between the meet expressions of Perm(n) elements as well as the meet operation on permutations itself can be reconstructed in terms of I-compatible adjacent pairs through these covering H6 intervals (see Fig. 4).
3. Recursive
constructions
of Perm(n)
First, it will be useful to get a representation of Perm(n) as a Galois lattice in order to investigate it with computer programs. Recall that for any lattice L, and for its sets J(L) of join and M(L) of meet-irreducible elements, the reduced representation of L as a Galois lattice (see Barbut and Monjardet, 1970) is given by L = L(J(L), M(L), s), and characterized by the isomorphism x E L /+ (J(L) l-l (Xl>M(L) l-lIx)>. Reciprocally, any relation R c X x Y defines the Galois lattice L(R, X, Y) = {(B ‘, B It) 1B C Y} through the Galois connection, which is set up by the decreasingmappingsA/+AT={yEYIaRyallaEA} (A5X)andBbB’= {x E X) xRb all b E B} (B C Y) (see Birkhoff, 1967, p. 124; Barbut and Monjardet, 1970; Markowsky, 1975; Wille, 1984). For defining a lattice, such a relation R will often be given through a table, whose rows and columns are indexed by X and Y.
i:4 1
14
l :1:34 Fig. 4. The meet-core of the permutation
lattice.
V. Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
81
Since each join-irreducible permutation y E J(Perm(n)) has only one adjacent pair that is T-compatible, it is the concatenation of two maximal blocks that are I-compatible, and can be written y = A : A (where ‘:’ separates the two complementary blocks). Notice that A is not an ideal of the linear order 1 < 2 < . . * < n, since y # 1. Dually, a meet-irreducible element p E M is formed of two maximal T-compatible blocks, and is written p = B : B. Hence, the relation (1, M, 5) can be derived by checking all the containments between the pI ( y ) and p,(p). Since it can become quickly intractable to deal with all those pairs to make the checking, a simpler test given in the following lemma will be most useful. Lemma
9. The order (J, M, 5) between the join- and meet-irreducible
elements of
Perm(n) is given by y 5 p e y = A : 2 and u = B : B satisfying either (1) n FA f~ B and (y - n) 5 ( /.L- n) in Perm(n - I ), where ( y - n) means erasing n in y, or (2) nEAfIfiandAnB=O. Proof. Case (1):
n $ A n B. The necessary condition is obvious when erasing n in the permutations y and CL.Conversely, let y ’ i 11’ be two join- and meet-irreducible elements in Perm(n - 1); adding II in the first block of p’ (second block of y ‘) does not increase the l-compatible pairs (T-compatible pairs for y’). Case (2): n E A II l?. First, suppose that A n B # 0 holds, so that there exists some x E A n B; this implies (x, n) Ep,( u) although (x, n)$p,(y), which contradicts y i p; hence, A fl B = 0 must hold. Second, suppose A n B = 0; let (x, y) E p,(p); since p = B : B is the concatenation of two maximal T-compatible blocks, this implies x E B and y E B; A n B = 0 implies x E A, so that (x, y) Ep,(y) holds true. 0
An even simpler characterization of (J, M, 5) has been given in Markowsky (1991): j$m iff min(A n i?) rmax(A n B). In the sequel, we will however make use of Lemma 9, since it leads to a simple recursive process that may be summarized as follows: Proposition 10. Starting from the 2 x 2 table of Perm(1 ), which is completely filled with crosses, the 2” X 2” table for Perm(n) (with lexicographically ordered rows1 columns) is obtained by copying four times the 2”-’ X 2”-’ table for Perm(n - 1) and erasing the crosses above the subtable diagonal for which n E A fl fi (see Fig.
5). The amount of redundancy is quite small (n + 1 extra rows/columns filled with X and therefore associated with the lattice O/l), as compared with the simplicity of this construction by mere duplication, which implies that the arrows defined below will also be obtained by duplication, without requiring complex checking. In Fig. 5, a non-reduced representation of Perm(n) is given for which the
82
V. Duquenne, A. Cherfouh I Mathematical Social Sciences 27 (1994) 73-89 C
0121
I
A:ii
E
F
G
z
:
:
2 21 1:.31**4 :332 3121
I 4
J
K
L
;;; 1:
x5,! 221. AI22 412
M
I
i 3
N
0
:4 4 4 3 3 3 2 121
01234 1 234
c=
2:134 12 34
e=
3:124
f=
13~24
g=
23:14 123
4
i= 4:123 j=
14:23
k=
24:13
1=
124:3
m=
34:12
n=
134~2
o=
234:l 1234
Fig. 5. A non-reduced table of Perm(5) with arrow relations (the join/meet labelled by lower-case/capital letters).
irreducible elements are
join-irreducible elements are labelled by lower-case letters; for c = 2: 134, A = 2 and A = 134. Dually, the columns of the corresponding meet-irreducible elements that are obtained by reverse images are labelled by capital letters: C = 431: 2, of which the blocks are B = 431 and B = 2. Our new graphic program GLAD (General Lattice Analysis and Design) has been used to produce understandable drawings of Perm(4) and Perm(5) and their meet cores (see Figs. 4, 7 and S), as well as to calculate the up/down arrows in Fig. 5, which define the weak perspectives between join- and meet-irreducible elements (see Crawley and Dilworth, 1973; Wille, 1983, 1984). Let us denote by y x ,u the fact that the entry (y, CL)is ‘ X ’ in Fig. 5, which means that y 5 p in Perm(l), while 7’~ means that /.L is a maximal meetirreducible which is not above y (dually for yL E.L).For p E M(Perm(n)), let CL*be its unique upper cover in Perm(n), and dually let y * be the unique lower cover of y E J(Perm(n)).
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V :V
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V. Duquenne, A. Cherfouh I MathematicalSocial Sciences 27 (1994) 73-89
The dual property holds for the T arrows in the column of k. The last point of (2) is proved like (1). 0 These two hereditary duplication properties of arrows are represented in Fig. 6 (see subtables l-211-3 and 4, respectively). On the other hand, as noticed in Le Conte de Poly-Barbut (1990), Perm(n) resembles a product of chains, which generalizes the generation of Boolean lattices by ‘doubling’. This can be made precise by the following Proposition 12, which expresses that perm(n) is constructible from C, X Perm(n - 1) by erasing some coverings regularly (see Figs. 2 and 7). Let 7, = (nl . . . (n - 1)) and p, = ((n - 1). . . In), and let C(n) be the number of cover relationships in Perm(n). Proposition
12. [A,) U (p,] G C, X Perm(n -I), and for any (YE (p,] [(u, ff v A,] z C,, holds. Let /3 s p,, reverse (x, y) in a; [(u, (Y v h,] has one element not
covered
by its join with /3, obtained
from
(Y by moving
n between
x and y. Hence,
C(n) = (n - 1) X (C(n - 1) + (n - l)!) = (n - 1) X n! /2. Proof.
The two first claims follow from the fact that (Yv A,, is constructed from (Y by moving n to the left in (n - 1) steps. Then y E [(Y, LXv A,] is not covered by y v p iff x and y are not neighbors in y. This proves C(n) = (n - 1) x C(n - 1) + (n - l)! X (n - l), by first counting all the cover relationships that are perspective to the cover relationships of the elements of (p,], and then all those which are not. The second formula is implied by the fact that all the elements of Perm(n) have (n - 1) upper and lower covers. Cl
Fig. 7. The meet core of Perm(5).
V. Duquenne, A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
85
Fig. 8. The congruence ‘sharing the same complements’.
4. The congruence
lattice of Perm(n)
Exploring congruences of Perm(n) should be useful to get a better insight of its structure: this requires studying the heredity of collapsing the elements. Thus, O(F, T) being the smallest congruence collapsing F and T, the perspectivities in Perm(3) imply _LO(F, T)C and EO(F, T)T (see Fig. 9). Mastering all these heredities calls for characterizing the congruence lattice C(Perm(n)), which is - as for any lattice (see Birkhoff, 1967, p. 138) -distributive, and therefore determined by the order of its meet-irreducible elements. (M(C(Perm(3))), 5) can thus be viewed as the union of two complete binary trees (having all their leaves at the same depth), rooted at the maxima O(C, T) and O(F, T). For n={l,...,n} and Perm(n) rooted at (1. . . n), let AA(n)= (M(C(Perm(n))). For any coatom (n . . . k(k + 1). . . l), let O((n . . . k)(k + 1) l), T) be the smallest congruence collapsing it with T, and let $k = {(YE i(n) ) cy 5 @((n.. . k(k + 1). . . l), T)}. Then, for k = 1, 2, . . . ,n-1, let S$= 9,\U { 9, ) k < j< n} and JZ~= $,\U {_ai) 1~ j< k} be what remains in $, after suppressing the other ideal elements ‘at right’ and ‘at left’. M(C(Perm(4))) is represented in Fig. 10, for which $!I = {C}, Se2= {F, E, G}, and Zn3= {L, J, N, I, M, K, 0} that are downward binary trees of depth 0, 1, and 2, respectively, and symmetrically for .& = {L}, -fE;= {F, 1, N}, and _rP = {C, E, G, I, M, K, 01. Now,
it is known that the weak perspectivities between join- and meetirreducible elements reflect the hereditary properties of collapsing the cover
86
K Duquenne,
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I Mathematical Social Sciences 27 (1994) 73-89
~=321 !..-..
@(C,T)
*/O\ 0(k)\
@(F,T)
/‘0cF.i
OG,CI/“\ 0.
@(E,F) .O
0&F)
0(W) . ... : ~=123 0(F,
T)={J!.%EFT}
MK(PermG!)))
C(Perm(3)) Fig. 9.
relationships when forming congruences (see Crawley and Dilworth, 1973, ch. lo), up to the point that the congruence lattice C(L), for a lattice L, is anti-isomorphic to the order of closed subsets of the bipartite graph of arrows on J(L) I”JM(L) (see Wille, 1983, section 3) that are subsets C for which there is no x in C and y not in C with x’y or xsy. Theorem 13. (A(n), 5) = o(5Bk 1k = 1.. . n - 1) U o(-r;P, ) k = 1.. . n - l), where n - 1 the S%$ [ resp. Zk] are complete binary trees of depth k - 1 for k=l... [n - k - 11, such that elements of Si cover two elements of .5%!i!i,, [Tj] and consist of all rank n - i - 1 [ j - l] elements in the & [&T&l. Proof. The property
is true for n = 3. Suppose that it holds remains to prove that it is also true for n. Claim 1. %!n_l is a complete binary tree of depth n v = (n - l)n(n - 2). . . 1 E M(Perm(n)) and x E J(Perm(n)) which is such that x $ v (see Figs. 5 and 6, for which x
Fig. 10. The meet-irreducible
elements
of C(Perm(4))
(each
for arbitrary n - 1; it 2 in (A(n), 5). Let be its reverse image, = 1 and v = L). Let
@(p, /.L*) is labelled
by p).
V. ~~q~~n~e, A. C~erfo~~ ( ~ut~e~aticaf Sociaf Sciences 27 (1994)
73-89
87
x = (y
E ~(P~~(~)) 1y 5 v} ) and dually let N = { ,u E ~~Perrn(~)) 1p 2 x}. By Propositions 10 and 11(2), the closed subset generated by x and Y is equal to {x, V} since the row of x and the column of ZJ contain a unique ’ and $, respectively; moreover, any element of X U M{ ,y, v} generates a closed subset which contains {,y, Y}, so that S(Y, T) is maximal in (92e_i, 5). Let now VI = (n - l)(n - 2)n.. . 1 and V, = (n - 2>n(n - 1). . . 1 (see Figs. 5 and 6 where or = J and V, = N). By Propositions 10 and 11 similarly, the closed subsets generated by V~and 9 are both minimal within the closed subsets generated by the elements of X U M{ x, Y}, so that their congruences are lower covers of O( V, T) in the order (S!&_, , 5). By induction on the elements of SR,_1 and by using the same argument at each step, (St,_ z, 5) is proved to be a binary tree of length rz - 2, since the step i generates 2’ elements, INI = 2”-r - 1, while c7;2i = 2”-’ - 1. Ciaim 2. In (J@(E), <)I each element of SY%!~_~ is an upper cover of two elements of 9$-l. This is a direct consequence of the second assertion of Proposition U(2) (see Fig. 10 and $Rz= {F, E, G}, whose elements are covering the pairs (J, n), (K O), and (1, MD. Claim 3. The elements of 9Zj ( j = 1. . . n-l) are of rank n-l-j in S” (k = 1.. . n - 1). Observe that the elements of 9$_, are the elements of rank 0 in all the z?$, k=l... n - 1. Hence, by using Claims 1 and 2 successively at each stepforj=l,..., 1~- I the claim holds finally. Cl
To understand these heredities, consider IF in Fig. lo- representing O(F, T) there-which is an upper cover of (E, G) in S?.2, and of (J, N) in 22,: this just
Fig. 11. The congruence lattice of Perm(4).
V. Duquenne,
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i Mathematical Social Sciences 27 (1994) 73-89
reflects that (E, T) is gluing two H6 intervals of maximum T, implying both collapsings (at left/right). Notice that all CLE M(Perm(n)) are perspective to atomslcoatoms of H6 intervals, and that the map i b IZ- i + 1 (i = 1, n) induces a symmetry in (A(n), 5). This explains how the hereditary properties of perspectivities in Perm(3), together with the shifts of hexagonal intervals according to Proposition 12, are responsible for the mixture of the double tree families in (J!(n), 5). Finally, since .&/u(n)has 2”-* minimal elements, Perm(n) is expressible as a subdirect product of as many irreducible factors (see C(Perm(4)) that has been drawn in Fig. 11 thanks to GLAD). If all the permutation lattice mysteries are not clarified yet, we think that using standard tools of Lattice Theory has made it a bit more understandable: the main line has been forgetting the semantics of permutations as much as possible, and emphasizing local constraints to try approaching more global properties.
Acknowledgements
This paper has been developing the m&moire de DEA passed by the second author in 1991, for which Pr. B. Monjardet is kindly thanked for his advice and amendments. Thanks are due to the referees for their great attention, and to la Maison des Sciences de 1’Homme and la Maison Suger for providing the facilities for writing the program GLAD. Last, a special appreciation is due to the team Ge’nttique Neuroghze’tique et Comportement for making us welcome.
References M. Barbut
and B. Monjardet,
Ordre
et classification,
vols. 1 & 2 (Hachette,
Paris,
1970).
M.K. Bennett and G. Birkhoff, Two families of Newman lattices, to appear in Algebra Universalis (1992). G. Birkhoff, Lattice Theory, 3rd edn. (AMS, Providence, 1967). C. Chaumeni Nembua, Permutokdre et choix social, These de MathCmatiques Appliqute aux Sciences Sociales, Univ. Paris (1989). C. Chaumeni Nembua and B. Monjardet, Les treillis pseudocomplCmentCs finis, Europ. J. Combinatorics 13 (1992) 89-107. P. Crawley and R.P. Dilworth, Algebraic Theory of Lattices (Prentice-Hall, Englewood Cliffs, N.J., 1973). V. Duquenne, Contextual implications between attributes and some representation properties for finite lattices, in: B. Ganter et al., eds., Beitrige zur Begriffsanalyse (Wissenschaftsverlag, Mannheim, 1987) 213-239. V. Duquenne, On the core of finite lattices, Discrete Math. 88 (1991) 133-147. G. Guilbaud and P. Rosenstiehl, Analyse algtbrique d’un scrutin, Math. Sci. Hum. 4 (1963) 9-33. C. Le Conte de Poly-Barbut, Le diagramme du treillis permutokdre est intersection des diagrammes de 2 produits directs d’ordres totaux, Math. Sci. Hum. 111 (1990) 49-53. G. Markowsky, The factorization and representation of lattices, Trans. Amer. Math. Sot. 203 (1975) 185-200.
V. Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
89
G. Markowsky, Permutation Lattices Revisited, Technical report, Univ. of Maine (1991). R. Wille, Subdirect decomposition of concept lattices, Algebra Universalis 17 (1983) 275-287. R. Wille, Liniendiagramme hierarchischer Begriffs-systeme, in: Anwendungen der Klassifikation: Datenanalyse und numerische Klassifikation (Indeks Verlag, Frankfurt, 1984) 32-51. T. Yanagimoto and M. Okamoto, Partial orderings of permutations and monotonicity of a rank correlation statistic, Ann. Inst. Statist. Math. 21 (1969) 4899506.
fQ 1994 - Elsevier
73
Science
B.V. All rights
reserved
On permutation lattices Vincent Duquenne” URA, CNRS 1294, G&attique, Neurogt%tique 75006 Paris, France
Ameziane
et Comportement,
45 Rue des Saints P&es,
Cherfouh
VFR de Mathtmatiques,
Vniversitt de Paris V, 12 Rue Cujas, 75005 Paris, France
Communicated by M.F. Janowitz Received 3 February 1992 Revised
14 August
1993
Abstract The lattice perm(n) of permutations of an n-element set n = (1,. . , n}, ‘rooted’ at (1,. , n), is shown to be meet- and join-semidistributive, which implies known results such as the non-existence of M,-sublattices, and that the complementation defines a congruence relation in Perm(n) with 2”-’ classes. The meet-core of Perm(n) is shown to be the set of meet-irreducible elements together with all the elements that have two upper covers that moreover generate a covering sublattice isomorphic to Perm(3); this shows that the meet operation is completely expressible in terms of reversing adjacent pairs. A recursive construction of Perm(n) as a Galois lattice -via a kind of summation process - is given, which has been a key for obtaining clear drawings of Perm(4) and Perm(5) with our new PC/VGA graphic program GLAD (General Lattice Analysis and Design). Finally, the congruence lattice C(Perm(n)) is recursively characterized through the order of its meet-irreducible elements. Key words:
Permutation;
of collapsing;
Lattices
semidistributivity;
Meet/ join core;
Galois
lattice
representation;
of congruences
List of symbols n = (1,. , . ) n}
a finite set permutations I-compatible pairs cardinal of. , . cover, order relations set operations lattice operations
a, P, Y . . f p,(a) I I -<,
<, i
C, 5 u, n, u, v, A, v, A * Corresponding
SSDI
n
author
0165-4896(93)00733-B
at: CNRS,
10-b rue A. Payen,
F75015
Paris,
France.
Heredity
V. Duquenne,
74
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
E, CL)
meet-essential elements meet-core a decreasing mapping subsets of C(Perm(n)) disjoint union (weak) perspectivities congruence
K,(L) A b
A’ = { y E Y ( aRy all a E A}
“H(n), Y&, ?i&. . . fi t ’ LT 5,
@(my m)
0. Introduction Permutations on a set play a prominent role in many fields of mathematics, as well as more applied topics like statistics, data analysis, up to experimental designs and genetics, or kinship systems in anthropology. The first property which is emphasized in every book is that starting from any permutation I = (123. . .) on a set n={1,2,..., n}, any permutation can be obtained by a series of transpositions reversing neighbors (see Fig. 1). This gives rise to the symmetric group S, on an n element set. It was observed long ago in the context of Social Choice (See Guilbaud and Rosenstiehl, 1963, and Barbut and Monjardet, 1970) that the transpositions also define the cover relation of a lattice, which is denoted by Perm(n). If the group-theoretic properties of permutations are well known, some lattice properties of Perm(n) are not so clarified yet. This paper aims at contributing to fill the gap and to the revival of interest in permutation lattices (Chaumeni Nembua, 1989; Le Conte de Poly-Barbut, 1990; Markowsky, 1991; Bennett and Birkhoff, 1992).
1. Lattice properties
of Perm(n)
Let n = {1,2, . . . , n}, Perm(n) be the set of all permutations on n, I = (123. . .) and T = (. . .321) be elements of Perm(n). For (YE Perm(n), let p,(n) 321
231 /ok0
21:\/
312
I
I 0
’ 12
12
132
23
I$ Fig.
1. Perm(3)
is ‘rooted’
at 123. The edges
are labelled
by the neighboring
locations.
V. Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994)
75
73-89
be the set of ordered pairs of distinct elements that are I-compatible in (Y,i.e. in the same order as in I. T-compatible ordered pairs and pT(o) are defined dually. Two ordered pairs of neighbors (i.e. elements at neighboring locations in the permutation) are adjacent if they share some element. It is often worthwhile viewing a permutation simply as a succession of maximal blocks of either I- or T-compatible adjacent pairs. p,(o) and pT(cx) are subsets of p,(l) and pT(T), respectively. Since any ordered pair (x, y) E n x n is either I-compatible or T-compatible, the duality between pT(o) and p,(o) may be stated as follows: PET
(*)
and
= PT(T)WPI(~))
p,(o)
= pI(WJWp,(4)
7
where Tr(R) stands for the transpose of a relation R. When p is obtained by reversing a I-compatible pair of neighbors in (Y, we write LY--< /3: let 5 be the transitive closure of the relation ‘--< or =‘. Thus, (Perm(n), 5) is the order of all permutations of n rooted at I, and is ranked since : foranycu c/3, ff --
(**>
iff /p,(o)]
= Ip,(p)]
-1.
Generalizations of this construction are studied at length in Bennett and Birkhoff (1992), while algorithmic aspects of Perm(n) are developed in Markowsky (1991). The set M c Perm(n) of permutations that have only one upper cover is characterized as follows: each p E M has exactly one I-compatible pair (x, y) of neighbors. The set J of permutations having only one lower cover is characterized dually. A lattice is meet-semidistributive (SD,,) if x A a = x A b 3 x A a = x A (a v b). Join-semidistributive lattices (SD,) are defined dually. R* is the transitive closure of a relation R. For standard notions on lattices, see Birkhoff (1967), Barbut and Monjardet (1970), or Crawley and Dilworth (1973). Proofs that (Perm(n), 5) is a lattice have been given in Guilbaud and Rosenstiehl (1963), Yanagimoto and Okamoto (1969), and in the more algorithmic construction given recently in Markowsky (1991): Theorem p,(o)
1. (Perm(n),
uPI(P
5) is a ranked compEemented lattice such that pI(o
and ~,(a
v P) = (~~(4
UPAP))*
A p)
=
hold.
Lemma 2. Let cy E Perm(n) and (Y+ = v { a0 E Perm(n) ( a0 >- a}. Then [a, (Y‘1 z X iEl Perm( IDi\) holds for the family (Bi);,, of maximal blocks made of I compatible adjacent pairs of (Y. Proof. Let cxi be obtained by reversing the blocks Bi of cy, for all i E Z. [a, ai] s Perm((B,[) holds, obviously. Let p E [ a, CY+] and pi= /3 A cxi (iEZ), so that p =‘ V iE, pi holds. Each pi is obtained from (Yby reversing the pairs of Bi that are also contained in pl(/3). Since Perm(n) is ranked, this implies in turn that
/? = V iEI pi holds, which shows that the mapping /3 * ( pi)igl is bijective; since it is, moreover, a 01-meet-preserving map [a, a+]--, Xi,,Perm(\B,(), this is a q lattice isomorphism, so that [(Y, (~‘1 z xiEI[ a, ai].
76
K Duquenne, A. Cherfouh I MathematicalSocial Sciences 27 (1994) 73-89
3241 3214
\I:
,‘-g:
1: 3,243
f/-l324
/
-c:
2314
2134
12w
Fig. 2. The permutation lattice Perm(l) ‘rooted’ at 1234 (V and A indicate join- and meet-irreducible elements, respectively).
In particular, Lemma 2 generalizes known properties (see Chaumeni Nembua, or [cx, cq v a2) E 1989): for LX-
3. (Perm(n),
5) is a meet- and join-semidistributive
lattice.
Proof.
Let x, (Y,p E Perm(n) be such that x A (Y= x A j? = x A cy A j3 ; in order to establish SD,, , it must be shown that: (*) x A cx = ,y A (a v /!I) holds. Claim 1. (*) holds when (YA p -< (Y, p. Let y be such that x A (Y --< y 5 x, and let CY,p be obtained from cx A j? by reversing (a, x) and (y, b), so that a < x 5 y < b holds. By Lemma 2, [LYA /3, (Y v /3] is isomorphic either to C, X C, when x # y, or to Perm(3) otherwise when x = y. Suppose that [(YA p, cy v ,6] = C, x C, holds; (Yv /3 is obtained from LYA p by reversing (a, X) as well as (Y, 6); hence P,(Y A (a v P)) = (P,(Y) U ~,(a v P))* = (P,(Y) U P~(” A P) ’ ((6 x)3 (Y, b)))* = ((P,(Y) U p,(a)) n (P,(Y) UP,(P)))* = (PI(Y A a) n P,(Y A P))* = P~(Y A a), since I~,(r)l = I(P~(Y A a)I - 1 impliesp,(y * Q’) =P,(Y) UP,(~) andp,(y A P) = P,(Y) Up,(P). Otherwise, when [a A P, (Yv P] z Perm(S), LYv p is obtained from LYA p by reversing the block (a X 6);
thus, P,(Y x), (x, b),
* (a ” P)) = (P,(Y) u PL(a ” P))* = (P,(Y) (a, b)})*; now, y >- y A ct’, y$cx, (Y transposing
(x, a)@pL(y), by transitivity, P>>* = /‘I(?’
so
that (a, x) Ep,(y); similarly, (x, b) up,, Hence, pI(y A (cy v p)) =
(a, b)Ep.(y).
A a A p).
u P,((y (a, x) in
A P) \ {(a,
(YA /3 imply which implies,
(p,(y)
U P~(‘Y A
V. Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
77
Claim 2. For x, (Y, /3 such that x A (Y= X A p there exists some y >- (Ywith y 5 LYv p and x A y = x A (YA p. Let (Y, and PI be such that (YA p --< CY,9 (Y and (YA p -< & i p and S, = [a A p, czl v PI]; then: either S, z C, X C, or S, z I-I6hold, and (*l) x A (Vs,) = x A (Y A j3 holds, by Claim 1; if vS, 2 (Y,then Claim 2 holds; otherwise, Q $ vS,, cxl 5 a and czl < Vs, implies (YA (VS,)# AS,; let CQ, & >- (YA (VS,) be such that (Y*5 (Y and &E S,. Let S, = [a A (VS,), cx2v p,]; (*l) and Claim 1 imply that similarly: (*2) x A (VS,) = ,y A cy A p holds. By reiterating this process taking each (Y~along a maximal chain of [a A /3, a], Claim 2 is reached by taking y >- a! in the first Si with V Si > (Y. By induction on the rank of y along a maximal chain of [(Y, (Yv p], the SD h law is proved. The SD, law is established dually. Cl
Many properties of Perm(n) are consequences of the SD laws, such is the absence of M,-sublattices (with 5 elements, 3 atoms). On the other hand, as observed for Perm(n) by Le Conte de Poly-Barbut (1990) and Markowsky (1991) (see below, Fig. 8), and more generally for pseudo-complemented lattices by Chaumeni Nembua (1989)) Chaumeni Nembua and Monjardet (1992) one obtains: Proposition 4. Let L be a complemented SD, and SD, lattice, A and C be its sets of atoms and coatoms, and let 0 be the relation ‘sharing a complement’. 0 is a lattice congruence relation on L having 2 IAt= 2”’ classes, of which the minimal
(maximal) elements are the joins of atoms (the meet of coatoms, respectively).
2. The core of the permutation
lattice
The core of (arbitrary) finite lattices has been introduced in Duquenne (1987) for dealing with the following question: What minimum piece of information must be added to the order of meet-irreducible elements (M(L), 5) of a lattice L in order to be in a position to reconstruct it? For traditional classes of lattices (e.g. geometric, modular, join/meet semidistributive) the meet- and join-cores have been characterized (Duquenne, 1991), by strengthening lemmas that depend on the semimodular laws. In the case of the permutation lattice, as opposed to these classes, since semimodular laws are not available a tailor-made approach has to be developed. Let us first recall some definitions (see Duquenne, 1987, 1991): an element x E L\M(L) is said to be meet-essential when the following holds: (ME).
there exists some proper order filter X of [x)\(x) for which AX = x and such that for any Y G X, either A Y = x or A YE X holds
We say in short that the triple (x, X, a) is meet-essential when the condition ME holds for (x, X) and a >- x ‘is not captured’: a E [x&Y. Hence, x is meetessential iff such a triple (x, X, a) exists. For any y E L\(x], let yzx = A {z E x 1
V. Duquenne,
78
A. Cherfouh
I Mathematical Social Sciences 27 (1994)
73-89
z 2 y} be the (unique) smallest element of X greater than y. The set of meet-essential elements is denoted by E,(L) and the meet-core is defined by: K,(L) = M(L) U E,(L). Theorem [Duquenne, 19871. For a finite lattice L and PC L, the mapping z-+ [z) fl P (z E L) is an isomorphism L+ .9(P) onto the filter lattice of the partial meet-semilattice defined by P in L, iff P _> K,(L).
The following lemma is needed to study arbitrary cores: N,-M, Lemma [Duquenne, 19911. Let (x, X, a) be a meet-essential triple in L, and let x,, x,EminX for which x,#x,‘.a’X= r\{yEXlyra}. Then, {x,, x2, a, a’“} generates one of the sublattices shown in Fig. 3. Lemma
5. Let (x, X, a) be a meet-essential triple of SD, lattice. Then, a’x E {x, a, azX, x,, a v x1) z N5
min X holds, and moreover for any x1 E min x\{a’“}, holds.
Proof. In cases (i), (ii), (iv), (v) and (vi) of the N,-M, Lemma, the inequality x2 A x1 = x2 A a # x2 A (x1 v a) = x2 holds, contradicting the SD h law. Hence only case (iii) can occur in Perm(n). 0 Lemma 6. Zf an element x E L of an arbitrary lattice L has only two upper covers, x1, x2 >- x, which moreover belong to a sublattice H6 g Perm(3), then x is a meet-essential element in L.
Let y,, y, be such that x, < y, and x2 < y2 in this H6 sublattice. Let an d zl, z2 E X. We claim that z1 A z2 E X or z1 A z2 = x hold: if X=[Y~)~[Y,) zr, z2 2 y, or zl, z2 2 y,, this is trivial. Suppose therefore that z1 E [yI)\[y2) and (Z, A z2 P x, or z1 A ~2 E [~2)\[~1) hold; ( z1 A z2 jZ’X and z1 A z2 #x) iI@ieS z2 2 x1), since x has only these two upper covers in L; suppose that z1 A z2 2 x1 holds; then x2 < z2 and x1 % z1 A z2 < z2 imply x1 v x2 = y1 v y, 5 z2, contradictProof.
ing z2 E [Y&Y&
0
Theorem 7. A permutation x is meet-essential in Perm(n) a, b >- x, such that [x, a v b] is isomorphic to Perm(3).
iff it has two covers,
The sufficiency is implied by Lemma 6. Conversely, let (x, X, a) be a meet-essential triple. By Theorem 3 and Lemma 5 x -< a < a* E min X, and there exists at least one x1 $a’” in min X such that N = {x, x1, a, aIX, x1 v a} is a N5 sublattice, since (min Xl 2 2. Claim 1. There exists b >- x for which [x, b v a] G H6 and is covering in Perm(n) If x -< xi, take b = x1; this is true by Lemma 2. Otherwise, let b >- x be such that b < x1 ; the triple (x, X, b) is meet-essential and bIX = x, holds, so Proof.
V. Duquenne,
x,va=x
A, Cherfouh
va
I Mathematical Social Sciences 27 (1994)
:X
x
79
73-89
va
a:X
2X
f< *1/‘\./\, 8
.I/+=\
.,:
\
x2
..’ /.” .‘I
1 # x2
a
m *s ...I (2
,J
I cf.‘.. X
\
,,..,O ..’‘.
d” X
(ii)
Gil
Fig. 3.
-< b
80
V. Duquenne, A. Cherfouh I Mathematical Social Sciences 27 (1994) 73-89
contrary would also imply that two distinct covering H, sublattices element chain, which is impossible. q
share a 3
Corollary 8. A permutation is meet-essential in Perm(n) iff it has exactly two pairs of neighbors that are adjacent and I-compatible.
Hence, all the equalities between the meet expressions of Perm(n) elements as well as the meet operation on permutations itself can be reconstructed in terms of I-compatible adjacent pairs through these covering H6 intervals (see Fig. 4).
3. Recursive
constructions
of Perm(n)
First, it will be useful to get a representation of Perm(n) as a Galois lattice in order to investigate it with computer programs. Recall that for any lattice L, and for its sets J(L) of join and M(L) of meet-irreducible elements, the reduced representation of L as a Galois lattice (see Barbut and Monjardet, 1970) is given by L = L(J(L), M(L), s), and characterized by the isomorphism x E L /+ (J(L) l-l (Xl>M(L) l-lIx)>. Reciprocally, any relation R c X x Y defines the Galois lattice L(R, X, Y) = {(B ‘, B It) 1B C Y} through the Galois connection, which is set up by the decreasingmappingsA/+AT={yEYIaRyallaEA} (A5X)andBbB’= {x E X) xRb all b E B} (B C Y) (see Birkhoff, 1967, p. 124; Barbut and Monjardet, 1970; Markowsky, 1975; Wille, 1984). For defining a lattice, such a relation R will often be given through a table, whose rows and columns are indexed by X and Y.
i:4 1
14
l :1:34 Fig. 4. The meet-core of the permutation
lattice.
V. Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
81
Since each join-irreducible permutation y E J(Perm(n)) has only one adjacent pair that is T-compatible, it is the concatenation of two maximal blocks that are I-compatible, and can be written y = A : A (where ‘:’ separates the two complementary blocks). Notice that A is not an ideal of the linear order 1 < 2 < . . * < n, since y # 1. Dually, a meet-irreducible element p E M is formed of two maximal T-compatible blocks, and is written p = B : B. Hence, the relation (1, M, 5) can be derived by checking all the containments between the pI ( y ) and p,(p). Since it can become quickly intractable to deal with all those pairs to make the checking, a simpler test given in the following lemma will be most useful. Lemma
9. The order (J, M, 5) between the join- and meet-irreducible
elements of
Perm(n) is given by y 5 p e y = A : 2 and u = B : B satisfying either (1) n FA f~ B and (y - n) 5 ( /.L- n) in Perm(n - I ), where ( y - n) means erasing n in y, or (2) nEAfIfiandAnB=O. Proof. Case (1):
n $ A n B. The necessary condition is obvious when erasing n in the permutations y and CL.Conversely, let y ’ i 11’ be two join- and meet-irreducible elements in Perm(n - 1); adding II in the first block of p’ (second block of y ‘) does not increase the l-compatible pairs (T-compatible pairs for y’). Case (2): n E A II l?. First, suppose that A n B # 0 holds, so that there exists some x E A n B; this implies (x, n) Ep,( u) although (x, n)$p,(y), which contradicts y i p; hence, A fl B = 0 must hold. Second, suppose A n B = 0; let (x, y) E p,(p); since p = B : B is the concatenation of two maximal T-compatible blocks, this implies x E B and y E B; A n B = 0 implies x E A, so that (x, y) Ep,(y) holds true. 0
An even simpler characterization of (J, M, 5) has been given in Markowsky (1991): j$m iff min(A n i?) rmax(A n B). In the sequel, we will however make use of Lemma 9, since it leads to a simple recursive process that may be summarized as follows: Proposition 10. Starting from the 2 x 2 table of Perm(1 ), which is completely filled with crosses, the 2” X 2” table for Perm(n) (with lexicographically ordered rows1 columns) is obtained by copying four times the 2”-’ X 2”-’ table for Perm(n - 1) and erasing the crosses above the subtable diagonal for which n E A fl fi (see Fig.
5). The amount of redundancy is quite small (n + 1 extra rows/columns filled with X and therefore associated with the lattice O/l), as compared with the simplicity of this construction by mere duplication, which implies that the arrows defined below will also be obtained by duplication, without requiring complex checking. In Fig. 5, a non-reduced representation of Perm(n) is given for which the
82
V. Duquenne, A. Cherfouh I Mathematical Social Sciences 27 (1994) 73-89 C
0121
I
A:ii
E
F
G
z
:
:
2 21 1:.31**4 :332 3121
I 4
J
K
L
;;; 1:
x5,! 221. AI22 412
M
I
i 3
N
0
:4 4 4 3 3 3 2 121
01234 1 234
c=
2:134 12 34
e=
3:124
f=
13~24
g=
23:14 123
4
i= 4:123 j=
14:23
k=
24:13
1=
124:3
m=
34:12
n=
134~2
o=
234:l 1234
Fig. 5. A non-reduced table of Perm(5) with arrow relations (the join/meet labelled by lower-case/capital letters).
irreducible elements are
join-irreducible elements are labelled by lower-case letters; for c = 2: 134, A = 2 and A = 134. Dually, the columns of the corresponding meet-irreducible elements that are obtained by reverse images are labelled by capital letters: C = 431: 2, of which the blocks are B = 431 and B = 2. Our new graphic program GLAD (General Lattice Analysis and Design) has been used to produce understandable drawings of Perm(4) and Perm(5) and their meet cores (see Figs. 4, 7 and S), as well as to calculate the up/down arrows in Fig. 5, which define the weak perspectives between join- and meet-irreducible elements (see Crawley and Dilworth, 1973; Wille, 1983, 1984). Let us denote by y x ,u the fact that the entry (y, CL)is ‘ X ’ in Fig. 5, which means that y 5 p in Perm(l), while 7’~ means that /.L is a maximal meetirreducible which is not above y (dually for yL E.L).For p E M(Perm(n)), let CL*be its unique upper cover in Perm(n), and dually let y * be the unique lower cover of y E J(Perm(n)).
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‘SPlOY [(u - wu -k) .dsa.d] (u - ?ql(u - /L) gz [rlk .&al] rllk uay ‘ploy 8 u v 3(1 -u) JO 8 u v 3 (1 - u) laqi?a 4 ‘X~pmJ f T/y4 k Xuv puv (1 . * . (z - u)u(l - u)) = d .doj-d,L ‘Qvnp puv ‘A.# d Icuv puv ((I - u)u(z - u) * . .I) = h _4oj rl tX : 8 ~~3uS11(z)~(~-~)~(u--)~~~~~83u~~ilvnp’.(u-~)~(u-X-)SST~~~ : V 3 U jj (I) ‘((U)LII.Iad)~ 3 8 : $$ = d PUV ((U)LKK& 3
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84
V. Duquenne, A. Cherfouh I MathematicalSocial Sciences 27 (1994) 73-89
The dual property holds for the T arrows in the column of k. The last point of (2) is proved like (1). 0 These two hereditary duplication properties of arrows are represented in Fig. 6 (see subtables l-211-3 and 4, respectively). On the other hand, as noticed in Le Conte de Poly-Barbut (1990), Perm(n) resembles a product of chains, which generalizes the generation of Boolean lattices by ‘doubling’. This can be made precise by the following Proposition 12, which expresses that perm(n) is constructible from C, X Perm(n - 1) by erasing some coverings regularly (see Figs. 2 and 7). Let 7, = (nl . . . (n - 1)) and p, = ((n - 1). . . In), and let C(n) be the number of cover relationships in Perm(n). Proposition
12. [A,) U (p,] G C, X Perm(n -I), and for any (YE (p,] [(u, ff v A,] z C,, holds. Let /3 s p,, reverse (x, y) in a; [(u, (Y v h,] has one element not
covered
by its join with /3, obtained
from
(Y by moving
n between
x and y. Hence,
C(n) = (n - 1) X (C(n - 1) + (n - l)!) = (n - 1) X n! /2. Proof.
The two first claims follow from the fact that (Yv A,, is constructed from (Y by moving n to the left in (n - 1) steps. Then y E [(Y, LXv A,] is not covered by y v p iff x and y are not neighbors in y. This proves C(n) = (n - 1) x C(n - 1) + (n - l)! X (n - l), by first counting all the cover relationships that are perspective to the cover relationships of the elements of (p,], and then all those which are not. The second formula is implied by the fact that all the elements of Perm(n) have (n - 1) upper and lower covers. Cl
Fig. 7. The meet core of Perm(5).
V. Duquenne, A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
85
Fig. 8. The congruence ‘sharing the same complements’.
4. The congruence
lattice of Perm(n)
Exploring congruences of Perm(n) should be useful to get a better insight of its structure: this requires studying the heredity of collapsing the elements. Thus, O(F, T) being the smallest congruence collapsing F and T, the perspectivities in Perm(3) imply _LO(F, T)C and EO(F, T)T (see Fig. 9). Mastering all these heredities calls for characterizing the congruence lattice C(Perm(n)), which is - as for any lattice (see Birkhoff, 1967, p. 138) -distributive, and therefore determined by the order of its meet-irreducible elements. (M(C(Perm(3))), 5) can thus be viewed as the union of two complete binary trees (having all their leaves at the same depth), rooted at the maxima O(C, T) and O(F, T). For n={l,...,n} and Perm(n) rooted at (1. . . n), let AA(n)= (M(C(Perm(n))). For any coatom (n . . . k(k + 1). . . l), let O((n . . . k)(k + 1) l), T) be the smallest congruence collapsing it with T, and let $k = {(YE i(n) ) cy 5 @((n.. . k(k + 1). . . l), T)}. Then, for k = 1, 2, . . . ,n-1, let S$= 9,\U { 9, ) k < j< n} and JZ~= $,\U {_ai) 1~ j< k} be what remains in $, after suppressing the other ideal elements ‘at right’ and ‘at left’. M(C(Perm(4))) is represented in Fig. 10, for which $!I = {C}, Se2= {F, E, G}, and Zn3= {L, J, N, I, M, K, 0} that are downward binary trees of depth 0, 1, and 2, respectively, and symmetrically for .& = {L}, -fE;= {F, 1, N}, and _rP = {C, E, G, I, M, K, 01. Now,
it is known that the weak perspectivities between join- and meetirreducible elements reflect the hereditary properties of collapsing the cover
86
K Duquenne,
A. Cherfouh
I Mathematical Social Sciences 27 (1994) 73-89
~=321 !..-..
@(C,T)
*/O\ 0(k)\
@(F,T)
/‘0cF.i
OG,CI/“\ 0.
@(E,F) .O
0&F)
0(W) . ... : ~=123 0(F,
T)={J!.%EFT}
MK(PermG!)))
C(Perm(3)) Fig. 9.
relationships when forming congruences (see Crawley and Dilworth, 1973, ch. lo), up to the point that the congruence lattice C(L), for a lattice L, is anti-isomorphic to the order of closed subsets of the bipartite graph of arrows on J(L) I”JM(L) (see Wille, 1983, section 3) that are subsets C for which there is no x in C and y not in C with x’y or xsy. Theorem 13. (A(n), 5) = o(5Bk 1k = 1.. . n - 1) U o(-r;P, ) k = 1.. . n - l), where n - 1 the S%$ [ resp. Zk] are complete binary trees of depth k - 1 for k=l... [n - k - 11, such that elements of Si cover two elements of .5%!i!i,, [Tj] and consist of all rank n - i - 1 [ j - l] elements in the & [&T&l. Proof. The property
is true for n = 3. Suppose that it holds remains to prove that it is also true for n. Claim 1. %!n_l is a complete binary tree of depth n v = (n - l)n(n - 2). . . 1 E M(Perm(n)) and x E J(Perm(n)) which is such that x $ v (see Figs. 5 and 6, for which x
Fig. 10. The meet-irreducible
elements
of C(Perm(4))
(each
for arbitrary n - 1; it 2 in (A(n), 5). Let be its reverse image, = 1 and v = L). Let
@(p, /.L*) is labelled
by p).
V. ~~q~~n~e, A. C~erfo~~ ( ~ut~e~aticaf Sociaf Sciences 27 (1994)
73-89
87
x = (y
E ~(P~~(~)) 1y 5 v} ) and dually let N = { ,u E ~~Perrn(~)) 1p 2 x}. By Propositions 10 and 11(2), the closed subset generated by x and Y is equal to {x, V} since the row of x and the column of ZJ contain a unique ’ and $, respectively; moreover, any element of X U M{ ,y, v} generates a closed subset which contains {,y, Y}, so that S(Y, T) is maximal in (92e_i, 5). Let now VI = (n - l)(n - 2)n.. . 1 and V, = (n - 2>n(n - 1). . . 1 (see Figs. 5 and 6 where or = J and V, = N). By Propositions 10 and 11 similarly, the closed subsets generated by V~and 9 are both minimal within the closed subsets generated by the elements of X U M{ x, Y}, so that their congruences are lower covers of O( V, T) in the order (S!&_, , 5). By induction on the elements of SR,_1 and by using the same argument at each step, (St,_ z, 5) is proved to be a binary tree of length rz - 2, since the step i generates 2’ elements, INI = 2”-r - 1, while c7;2i = 2”-’ - 1. Ciaim 2. In (J@(E), <)I each element of SY%!~_~ is an upper cover of two elements of 9$-l. This is a direct consequence of the second assertion of Proposition U(2) (see Fig. 10 and $Rz= {F, E, G}, whose elements are covering the pairs (J, n), (K O), and (1, MD. Claim 3. The elements of 9Zj ( j = 1. . . n-l) are of rank n-l-j in S” (k = 1.. . n - 1). Observe that the elements of 9$_, are the elements of rank 0 in all the z?$, k=l... n - 1. Hence, by using Claims 1 and 2 successively at each stepforj=l,..., 1~- I the claim holds finally. Cl
To understand these heredities, consider IF in Fig. lo- representing O(F, T) there-which is an upper cover of (E, G) in S?.2, and of (J, N) in 22,: this just
Fig. 11. The congruence lattice of Perm(4).
V. Duquenne,
88
A. Cherfouh
i Mathematical Social Sciences 27 (1994) 73-89
reflects that (E, T) is gluing two H6 intervals of maximum T, implying both collapsings (at left/right). Notice that all CLE M(Perm(n)) are perspective to atomslcoatoms of H6 intervals, and that the map i b IZ- i + 1 (i = 1, n) induces a symmetry in (A(n), 5). This explains how the hereditary properties of perspectivities in Perm(3), together with the shifts of hexagonal intervals according to Proposition 12, are responsible for the mixture of the double tree families in (J!(n), 5). Finally, since .&/u(n)has 2”-* minimal elements, Perm(n) is expressible as a subdirect product of as many irreducible factors (see C(Perm(4)) that has been drawn in Fig. 11 thanks to GLAD). If all the permutation lattice mysteries are not clarified yet, we think that using standard tools of Lattice Theory has made it a bit more understandable: the main line has been forgetting the semantics of permutations as much as possible, and emphasizing local constraints to try approaching more global properties.
Acknowledgements
This paper has been developing the m&moire de DEA passed by the second author in 1991, for which Pr. B. Monjardet is kindly thanked for his advice and amendments. Thanks are due to the referees for their great attention, and to la Maison des Sciences de 1’Homme and la Maison Suger for providing the facilities for writing the program GLAD. Last, a special appreciation is due to the team Ge’nttique Neuroghze’tique et Comportement for making us welcome.
References M. Barbut
and B. Monjardet,
Ordre
et classification,
vols. 1 & 2 (Hachette,
Paris,
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M.K. Bennett and G. Birkhoff, Two families of Newman lattices, to appear in Algebra Universalis (1992). G. Birkhoff, Lattice Theory, 3rd edn. (AMS, Providence, 1967). C. Chaumeni Nembua, Permutokdre et choix social, These de MathCmatiques Appliqute aux Sciences Sociales, Univ. Paris (1989). C. Chaumeni Nembua and B. Monjardet, Les treillis pseudocomplCmentCs finis, Europ. J. Combinatorics 13 (1992) 89-107. P. Crawley and R.P. Dilworth, Algebraic Theory of Lattices (Prentice-Hall, Englewood Cliffs, N.J., 1973). V. Duquenne, Contextual implications between attributes and some representation properties for finite lattices, in: B. Ganter et al., eds., Beitrige zur Begriffsanalyse (Wissenschaftsverlag, Mannheim, 1987) 213-239. V. Duquenne, On the core of finite lattices, Discrete Math. 88 (1991) 133-147. G. Guilbaud and P. Rosenstiehl, Analyse algtbrique d’un scrutin, Math. Sci. Hum. 4 (1963) 9-33. C. Le Conte de Poly-Barbut, Le diagramme du treillis permutokdre est intersection des diagrammes de 2 produits directs d’ordres totaux, Math. Sci. Hum. 111 (1990) 49-53. G. Markowsky, The factorization and representation of lattices, Trans. Amer. Math. Sot. 203 (1975) 185-200.
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G. Markowsky, Permutation Lattices Revisited, Technical report, Univ. of Maine (1991). R. Wille, Subdirect decomposition of concept lattices, Algebra Universalis 17 (1983) 275-287. R. Wille, Liniendiagramme hierarchischer Begriffs-systeme, in: Anwendungen der Klassifikation: Datenanalyse und numerische Klassifikation (Indeks Verlag, Frankfurt, 1984) 32-51. T. Yanagimoto and M. Okamoto, Partial orderings of permutations and monotonicity of a rank correlation statistic, Ann. Inst. Statist. Math. 21 (1969) 4899506.