The weak order on integer posets

0.81 MB·

by  Grégory Chatel

THE WEAK ORDER ON INTEGER POSETS
GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
Abstract. Theweakorderonthesymmetricgroupnaturallyextendstoalatticeonallinteger
binaryrelations. Wefirstshowthatthesubposetofthisweakorderinducedbyintegerposets
defines as well a lattice. We then study the subposets of this weak order induced by specific
families of integer posets corresponding to the elements, the intervals, and the faces of the
permutahedron,theassociahedron,andsomerecentgeneralizationsofthose.
The weak order is the lattice on the symmetric group S(n) defined as the inclusion order of
7 inversions, where an inversion of σ ∈S(n) is a pair of values a<b such that σ−1(a)>σ−1(b). It
1 is a fundamental tool for the study of the symmetric group, in connection to reduced expressions
0
of permutations as products of simple transpositions. It can also be seen as an orientation of the
2
skeletonofthepermutahedron(theconvexhullofallpermutationsofS(n)seenasvectorsinRn).

n The weak order naturally extends to all integer binary relations, i.e. binary relations on [n].
a
Namely, for any two integer binary relations R,S on [n], we define in this paper
J
7 R(cid:52)S ⇐⇒ RInc ⊇SInc and RDec ⊆SDec,
2
where RInc:= {(a,b)∈R|a≤b} and RDec:= {(b,a)∈R|a≤b} respectively denote the increas-
] inganddecreasingsubrelationsofR. Wecallthisordertheweak orderonintegerbinaryrelations,
O
see Figure 1. The central result of this paper is the following statement, see Figure 5.
C
Theorem 1. The weak order on the integer posets on [n] is a lattice.
.
h
Ourmotivationforthisresultisthatmanyrelevantcombinatorialobjectscanbeinterpretedby
t
a specificintegerposets,andthesubposetsoftheweakorderinducedbythesespecificintegerposets
m
often correspond to classical lattice structures on these combinatorial objects. To illustrate this,
[ westudyspecificintegerposetscorrespondingtotheelements,totheintervals,andtothefacesin

1 theclassicalweakorder,theTamariandCambrianlattices[MHPS12,Rea06],thebooleanlattice,
v and other related lattices defined in [PP16]. By this systematic approach, we rediscover and shed
5 light on lattice structures studied by G. Chatel and V. Pons on Tamari interval posets [CP15],
9
by G. Chatel and V. Pilaud on Cambrian and Schr¨oder-Cambrian trees [CP14], by D. Krob,
9
M. Latapy, J.-C. Novelli, H.-D. Phan and S. Schwer on pseudo-permutations [KLN+01], and by
7
0 P. Palacios and M. Ronco [PR06] and J.-C. Novelli and J.-Y. Thibon [NT06] on plane trees.
.
1
Part 1. The weak order on integer posets
0
7
1.1. The weak order on integer binary relations
1
:
v 1.1.1. Integer binary relations. Our main object of focus are binary relations on integers. An
i integer(binary)relationofsizenisabinaryrelationon[n]:={1,...,n},thatis,asubsetRof[n]2.
X
As usual, we write equivalently (u,v)∈R or uRv, and similarly, we write equivalently (u,v)∈/ R
ar or u(cid:54)Rv. Recall that a relation R∈[n]2 is called:
• reflexive if uRu for all u∈[n],
• transitive if uRv and vRw implies uRw for all u,v,w ∈[n],
• symmetric if uRv implies vRu for all u,v ∈[n],
• antisymmetric if uRv and vRu implies u=v for all u,v ∈[n].
From now on, we only consider reflexive relations. We denote by R(n) (resp. T(n), resp. S(n),
resp. A(n)) the collection of all reflexive (resp. reflexive and transitive, resp. reflexive and sym-
metric, resp.reflexiveandantisymmetric)integerrelationsofsizen. WedenotebyC(n)thesetof
integer congruencesofsizen, thatis, reflexivetransitivesymmetricintegerrelations, andbyP(n)
VPiwaspartiallysupportedbytheFrenchANRgrantSC3A(15CE40000401).
1
2 GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
thecollectionofintegerposetsofsizen,thatis,reflexivetransitiveantisymmetricintegerrelations.
In all these notations, we forget the n when we consider a relation without restriction on its size.
A subrelation of R ∈ R(n) is a relation S ∈ R(n) such that S ⊆ R as subsets of [n]2. We
say that S coarsens R and R extends S. The extension order defines a graded lattice structure
on R(n) whose meet and join are respectively given by intersection and union. The complemen-
tation R(cid:55)→{(u,v)|u=v or u(cid:54)Rv} is an antiautomorphism of (R(n),⊆,∩,∪) and makes it an
ortho-complemented lattice.
NotethatT(n),S(n)andA(n)areallstablebyintersection,whileonlyS(n)isstablebyunion.
Inotherwords,(S(n),⊆,∩,∪)isasublatticeof(R(n),⊆,∩,∪),while(T(n),⊆)and(A(n),⊆)are
meet-semisublattice of (R(n),⊆,∩) but not sublattices of (R(n),⊆,∩,∪). However, (T(n),⊆) is
a lattice. To see it, consider the transitive closure of a relation R∈R(n) defined by
Rtc:=(cid:8)(u,w)∈[n]2 (cid:12)(cid:12)∃v1,...,vp ∈[n] such that u=v1Rv2R...Rvp−1Rvp =w(cid:9).
ThetransitiveclosureRtc isthecoarsesttransitiverelationcontainingR. Itfollowsthat(T(n),⊆)
is a lattice where the meet of R,S ∈ R(n) is given by R∩S and the join of R,S ∈ R(n) is given
by (R∪S)tc. Since the transitive closure preserves symmetry, the subposet (C(n),⊆) of integer
congruences is a sublattice of (T(n),⊆).
1.1.2. Weak order. From now on, we consider both a relation R and the natural order < on [n]
simultaneously. To limit confusions, we try to stick to the following convention throughout the
paper. We use couples (u,v) when we do not know whether u<v or u>v for the natural order.
In contrast, we use couples (a,b) and (b,a) when we know that a≤b for the natural order.
Let In:= (cid:8)(a,b)∈[n]2 (cid:12)(cid:12)a≤b(cid:9) and Dn:= (cid:8)(b,a)∈[n]2 (cid:12)(cid:12)a≤b(cid:9). Observe that In∪Dn = [n]2
while I ∩D = {(a,a)|a∈[n]}. We say that the relation R ∈ R(n) is increasing (resp. de-
n n
creasing) when R ⊆ I (resp. R ⊆ D ). We denote by I(n) (resp. D(n)) the collection of all
n n
increasing (resp. decreasing) relations on [n]. The increasing and decreasing subrelations of an
integer relation R∈R(n) are the relations defined by:
RInc:=R∩In =(cid:8)(a,b)∈R(cid:12)(cid:12)a≤b(cid:9)∈I(n) and RDec :=R∩Dn =(cid:8)(b,a)∈R(cid:12)(cid:12)a≤b(cid:9)∈D(n).
Inourpictures,wealwaysrepresentanintegerrelationR∈R(n)asfollows: wewritethenumbers
1,...,nfromlefttorightandwedrawtheincreasingrelationsofRaboveinblueandthedecreasing
relations of R below in red. Although we only consider reflexive relations, we always omit the
relations (i,i) in the pictures (as well as in our explicit examples). See e.g. Figure 1.
Besides the extension lattice mentioned above in Section 1.1.1, there is another natural poset
structure on R(n), whose name will be justified in Section 2.1.
Definition2. The weakorderonR(n)istheorderdefinedbyR(cid:52)SifRInc ⊇SInc andRDec ⊆SDec.
The weak order on R(3) is illustrated in Figure 1. Observe that the weak order is obtained by
combiningtheextensionlatticeonincreasingsubrelationswiththecoarseninglatticeondecreasing
subrelations. Inotherwords,R(n)isthesquareofan(cid:0)n(cid:1)-dimensionalbooleanlattice. Itexplains
2
the following statement.
Proposition 3. The weak order (R(n),(cid:52)) is a graded lattice whose meet and join are given by
R∧ S=(RInc∪SInc)∪(RDec∩SDec) and R∨ S=(RInc∩SInc)∪(RDec∪SDec).
R R
Proof. Theweakorderisclearlyaposet(antisymmetrycomesfromthefactthatR=RInc∪RDec).
ItscoverrelationsarealloftheformR(cid:52)R(cid:114){(a,b)}foraRIncborR(cid:114){(b,a)}(cid:52)RwithbRDeca.
Therefore, the weak order is graded by R (cid:55)→ | RDec | − | RInc |. To check that it is a lattice,
consider R,S ∈ R(n). Observe first that R∧ S is indeed below both R and S in weak order.
R
Moreover,ifT(cid:52)RandT(cid:52)S,thenTInc ⊇RInc∪SInc andTDec ⊆RDec∩SDec,sothatT(cid:52)R∧ S.
R
This proves that R∧ S is indeed the meet of R and S. The proof is similar for the join. (cid:3)
R
Remark 4. DefinethereverseofarelationR∈RasRrev:= (cid:8)(u,v)∈[n]2 (cid:12)(cid:12)(v,u)∈R(cid:9). Observe
that (Rrev)Inc =(RDec)rev and (Rrev)Dec = (RInc)rev. Therefore, the reverse map R (cid:55)→ Rrev defines
an antiautomorphism of the weak order (R(n),(cid:52),∧ ,∨ ). Note that it preserves symmetry,
R R
antisymmetry and transitivity.
THE WEAK ORDER ON INTEGER POSETS 3
3
2
1
3
2
1
3 d.
2 te
3 3 t
1 mi
2 2
3 o
1 1 e
2 r
a
3 3
1 n]
2 2 [
1 3 1 ∈
2 i
3 3 3 3 r
1 o
2 2 2 2 f
1 1 23 1 1 i,i()
3 3
1 ns
2 2 o
1 3 1 ati
2 el
3 3 3 3 r
1
2 2 2 2 ve
1 1 3 1 1 xi
2 fle
3 3 e
1 r
2 3 2 All
1 1
3 3 2 3 3 3.
1 e
2 2 2 2 z
3 si
1 1 1 1 f
2 o
3 3 3 3 s
1 n
2 2 2 2 o
3 ti
1 1 1 1 a
2 el
3 3 3 3 r
1 y
2 2 2 2 r
1 1 3 1 1 na
2 bi
3 1 3 er
2 2 g
1 3 1 nte
2 i
3 3 3 3 )
1 e
2 2 3 2 2 xiv
1 1 1 1 e
2 efl
3 3 r
1 (
2 2 n
3 o
1 1
2 er
3 3 3 3 d
1 r
2 2 2 2 o
3 k
1 1 1 1 a
2 we
3 3
1 e
2 2 h
3 T
1 1
2 .
1
3 3
1 e
2 2 r
3 u
1 1 g
2 i
F
1
3
2
1
3
2
1
4 GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
1.2. The weak order on integer posets
In this section, we show that the three subposets of the weak order (R(n),(cid:52)) induced by
antisymmetric relations, by transitive relations, and by posets are all lattices (although the last
two are not sublattices of (R(n),(cid:52),∧ ,∨ )).
R R
1.2.1. Antisymmetric relations. We first treat the case of antisymmetric relations. Figure 2
shows the meet and join of two antisymmetric relations, and illustrates the following statement.
Proposition 5. The meet ∧ and the join ∨ both preserve antisymmetry. Thus, the antisym-
R R
metric relations induce a sublattice (A(n),(cid:52),∧ ,∨ ) of the weak order (R(n),(cid:52),∧ ,∨ ).
R R R R
Proof. Let R,S ∈ A(n). Let a < b ∈ [n] be such that (b,a) ∈ R∧ S. Since (b,a) is decreasing
R
and (R∧ S)Dec = RDec ∩SDec, we have bRDec a and bSDec a. By antisymmetry of R and S,
R
we obtain that a(cid:54)RInc b and a(cid:54)SInc b. Therefore, (a,b) ∈/ RInc ∪SInc = (R∧ S)Inc. We conclude
R
that (b,a) ∈ R∧ S implies (a,b) ∈/ R∧ S and thus that R∧ S is antisymetric. The proof is
R R R
identical for ∨ . (cid:3)
R
R∈A(4) S∈A(4) R∧ S∈A(4) R∨ S∈A(4)
R R
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Figure 2. The meet R∧ S and join R∨ S of two antisymmetric relations R,S.
R R
Our next two statements describe all cover relations in (A(n),(cid:52)).
Proposition 6. All cover relations in (A(n),(cid:52)) are cover relations in (R(n),(cid:52)). In particu-
lar, (A(n),(cid:52)) is still graded by R(cid:55)→|RDec|−|RInc|.
Proof. Consider a cover relation R(cid:52)S in (A(n),(cid:52)). We have RInc ⊇SInc and RDec ⊆SDec where
at least one of the inclusions is strict. Suppose first that RInc (cid:54)= SInc. Let (a,b)∈RInc(cid:114)SInc
andT:= R(cid:114){(a,b)}. NotethatTisstillantisymmetricasitisobtainedbyremovinganarcfrom
an antisymmetric relation. Moreover, we have R (cid:54)= T and R (cid:52) T (cid:52) S. Since S covers R, this
implies that S=T=R(cid:114){(a,b)}. We prove similarly that if RDec (cid:54)=SDec, there exists a<b such
that S=R∪{(b,a)}. In both cases, R(cid:52)S is a cover relation in (R(n),(cid:52)). (cid:3)
Corollary 7. In the weak order (A(n),(cid:52)), the antisymmetric relations that cover a given anti-
symmetric relation R∈A(n) are precisely the relations
• R(cid:114){(a,b)} for a<b such that aRb,
• R∪{(b,a)} for a<b such that a(cid:54)Rb and b(cid:54)Ra.
1.2.2. Transitive relations. We now consider transitive relations. Observe first that the sub-
poset (T(n),(cid:52)) of (R(n),(cid:52)) is not a sublattice since ∧ and ∨ do not preserve transitivity (see
R R
e.g. Figure 4). When R and S are transitive, we need to transform R∧ S to make it a transitive
R
relation R∧ S. We proceed in two steps described below.
T
Semitransitive relations Before dealing with transitive relations, we introduce the inter-
mediate notion of semitransitivity. We say that a relation R∈R is semitransitive when both RInc
andRDec aretransitive. WedenotebyST(n)thecollectionofallsemitransitiverelationsofsizen.
Figure 3 illustrates the following statement.
Proposition 8. The weak order (ST(n),(cid:52)) is a lattice whose meet and join are given by
R∧ S=(RInc∪SInc)tc∪(RDec∩SDec) and R∨ S=(RInc∩SInc)∪(RDec∪SDec)tc.
ST ST
Proof. Let R,S ∈ ST(n). Observe first that R∧ S is indeed semitransitive and below both R
ST
andS. Moreover,ifasemitransitiverelationTissuchthatT(cid:52)RandT(cid:52)S,thenTInc ⊇RInc∪SInc
andTDec ⊆RDec∩SDec. BysemitransitivityofT,wegetTInc ⊇(RInc∪SInc)tc,sothatT(cid:52)R∧ S.
ST
This proves that R∧ S is indeed the meet of R and S. The proof is similar for the join. (cid:3)
ST
THE WEAK ORDER ON INTEGER POSETS 5
R∈ST(4) S∈ST(4) R∧ S∈/ ST(4) R∧ S∈ST(4)
R ST
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Figure 3. Two semi-transitive relations R,S and their meets R∧ S and R∧ S.
R ST
As in the previous section, we describe all cover relations in (ST(n),(cid:52)).
Proposition 9. All cover relations in (ST(n),(cid:52)) are cover relations in (R(n),(cid:52)). In particular,
(ST(n),(cid:52)) is still graded by R(cid:55)→|RDec|−|RInc|.
Proof. ConsideracoverrelationR(cid:52)Sin(ST(n),(cid:52)). WehaveRInc ⊇SInc andRDec ⊆SDec where
at least one of the inclusions is strict. Suppose first that RInc (cid:54)= SInc. Let (a,b) ∈ RInc (cid:114)SInc be
such that b−a is minimal, and let T:= R(cid:114){(a,b)}. Observe that there is no a < i < b such
that aRiRb. Otherwise, by minimality of b−a, we would have aSi and iSb while a(cid:54)S b,
contradicting the transitivity of SInc. It follows that TInc is still transitive. Since TDec = RDec is
also transitive, we obtain that T is semitransitive. Moreover, we have R (cid:54)= T and R (cid:52) T (cid:52) S.
Since S covers R, this implies that S=T=R(cid:114){(a,b)}. We prove similarly that if RDec (cid:54)=SDec,
there exists (b,a) such that S = R∪{(b,a)}: in this case, one needs to pick (b,a) ∈ SDec(cid:114)RDec
with b−a maximal. In both cases, R(cid:52)S is a cover relation in (R(n),(cid:52)). (cid:3)
Corollary 10. In the weak order (ST(n),(cid:52)), the semitransitive relations that cover a given
semitransitive relation R∈ST(n) are precisely the relations
• R(cid:114){(a,b)} for a<b such that aRb and there is no a<i<b with aRiRb,
• R∪{(b,a)} for a<b such that b(cid:54)Ra and there is no i<a with aRi but b(cid:54)Ri and similarly
no b<j with jRb but j(cid:54)Ra.
Transitive relations We now consider transitive relations. Note that T(n) ⊆ ST(n) but
ST(n)(cid:54)⊆T(n). Inparticular, R∧ SandR∨ SmaynotbetransitiveevenifRandSare(see
ST ST
Figure 4). To see that the subposet of the weak order induced by transitive relations is indeed a
lattice, we therefore need operations which ensure transitivity and are compatible with the weak
order. For R∈R, define the transitive decreasing deletion of R as
Rtdd:=R(cid:114)(cid:8)(b,a)∈RDec (cid:12)(cid:12)∃i≤b and j ≥a such that iRbRaRj while i(cid:54)Rj(cid:9),
and the transitive increasing deletion of R as
Rtid:=R(cid:114)(cid:8)(a,b)∈RInc (cid:12)(cid:12)∃i≥a and j ≤b such that iRaRbRj while i(cid:54)Rj(cid:9).
Note that in these definitions, i and j may coincide with a and b (since we assumed that all
our relations are reflexive). Figure 4 illustrates the transitive decreasing deletion: the rightmost
relation R∧ S is indeed obtained as (R∧ S)tdd. Observe that two decreasing relations have
T ST
been deleted: (3,1) (take i=2 and j =1, or i=3 and j =2) and (4,1) (take i=4 and j =2).
Remark 11. The idea of the transitive decreasing deletion is to delete all decreasing relations
which prevent the binary relation to be transitive. It may thus seem more natural to assume in
thedefinitionofRtdd thateitheri=borj =a. However,thiswouldnotsufficetoruleoutallnon-
transitive relations, consider for example the relation [4]2 (cid:114){(2,3),(3,2)}. We would therefore
need to iterate the deletion process, which would require to prove a converging property. Our
definition of Rtdd simplifies the presentation as it requires only one deletion step.
Lemma 12. For any relation R∈R, we have Rtdd (cid:52)R(cid:52)Rtid.
Proof. Rtdd is obtained from R by deleting decreasing relations. Therefore (Rtdd)Inc = RInc and
(Rtdd)Dec ⊆RDec and thus Rtdd (cid:52) R by definition of the weak order. The argument is similar
for Rtid. (cid:3)
6 GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
R∈T(4) S∈T(4) R∧ S∈/ ST(4) R∧ S∈ST(4)\T(4) R∧ S∈T(4)
R ST T
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Figure 4. Two transitive relations R,S and their meets R∧ S, R∧ S and R∧ S.
R ST T
Lemma 13. If R∈R is semitransitive, then Rtdd and Rtid are transitive.
Proof. We prove the result for Rtdd, the proof being symmetric for Rtid. Set
U:= (cid:8)(b,a)∈RDec (cid:12)(cid:12)∃i≤b and j ≥a such that iRbRaRj while i(cid:54)Rj(cid:9),
so that Rtdd = R(cid:114)U with (Rtdd)Inc = RInc and (Rtdd)Dec = RDec (cid:114)U. Let u,v,w ∈ [n] be such
that uRtddv and vRtddw. We want to prove that uRtddw. We distinguish six cases according to
the relative order of u,v,w:
(i) Ifu<v <w,thenuRIncvandvRIncw. ThusuRIncwbytransitivityofRInc,andthusuRtddw.
(ii) If u<w <v, then uRIncv and vRDecw. Since v(cid:54)Uw, we have uRIncw and thus uRtddw.
(iii) If v <u<w, then uRDecv and vRIncw. Since u(cid:54)Uv, we have uRIncw and thus uRtddw.
(iv) If v < w < u, then uRDec v and v RInc w. Since u(cid:54)U v, we have uRDec w. Assume by
contradiction that uUw. Then there is i ≤ u and j ≥ w such that iRuRwRj but i(cid:54)Rj.
Since vRIncw and wRIncj, the transitivity of RInc ensures that vRj. We obtain that uUv,
a contradiction. Therefore, u(cid:54)Uw and uRtddw.
(v) If w < u < v, then uRInc v and v RDec w. Since v(cid:54)U w, we have uRDec w. Assume by
contradiction that uUw. Then there is i ≤ u and j ≥ w such that iRuRwRj but i(cid:54)Rj.
Since iRIncu and uRIncv, the transitivity of RInc ensures that iRv. We obtain that vUw,
a contradiction. Therefore, u(cid:54)Uw and uRtddw.
(vi) If w <v <u, then uRDecv and vRDecw, so that uRDecw by transitivity of RDec. Assume
by contradiction that uUw. Then there is i≤u and j ≥w such that iRuRwRj but i(cid:54)Rj.
Since u(cid:54)Uv and v(cid:54)Uw, we obtain that iRv and v Rj. If i ≤ v, then we have i ≤ v and
j ≥w with iRvRwRj and i(cid:54)Rj contradicting the fact that v(cid:54)Uw. Similarly, if j ≥v, we
have i≤u and j ≥v with iRuRvRj and i(cid:54)Rj contradicting the fact that u(cid:54)Uv. Finally,
if j <v <i, we have iRDecvRDecj and i(cid:54)RDecj contradicting the transitivity of RDec. (cid:3)
Remark 14. We observed earlier that the transitive closure Rtc is the coarsest transitive re-
lation containing R. For R ∈ ST, Lemmas 12 and 13 show that Rtdd is a transitive rela-
tion below R in weak order. However, there might be other transitive relations S with S (cid:52) R
and which are not comparable to Rtdd in weak order. For example, consider R:={(1,3),(3,2)}
and S:={(1,2),(1,3),(3,2)}. Then S is transitive and S (cid:52) R while S is incomparable to Rtdd =
{(1,3)} in weak order.
We use the maps R (cid:55)→ Rtdd and R (cid:55)→ Rtid to obtain the main result of this section. Figure 4
illustrates all steps of a meet computation in T(4).
Proposition 15. The weak order (T(n),(cid:52)) is a lattice whose meet and join are given by
R∧ S=(cid:0)(RInc∪SInc)tc∪(RDec∩SDec)(cid:1)tdd and R∨ S=(cid:0)(RInc∩SInc)∪(RDec∪SDec)tc(cid:1)tid.
T T
Proof. The weak order (T(n),(cid:52)) is a subposet of (R(n),(cid:52)). It is also clearly bounded: the weak
order minimal transitive relation is In = (cid:8)(a,b)∈[n]2 (cid:12)(cid:12)a≤b(cid:9) while the weak order maximal
transitive relation is Dn = (cid:8)(b,a)∈[n]2 (cid:12)(cid:12)a≤b(cid:9). Therefore, we only have to show that any two
transitive relations admit a meet and a join. We prove the result for the meet, the proof for the
join being symmetric.
LetR,S∈T(n)andM:=R∧ST S=(RInc∪SInc)tc∪(RDec∩SDec), sothatR∧T S=Mtdd. First
we have M(cid:52)R so that R∧ S=Mtdd (cid:52)M(cid:52)R by Lemma 12. Similarly, R∧ S(cid:52)S. Moreover,
T T
R∧ S is transitive by Lemma 13. It thus remains to show that R∧ S is larger than any other
T T
transitive relation smaller than both R and S.
THE WEAK ORDER ON INTEGER POSETS 7
Consider thus another transitive relation T ∈ T(n) such that T (cid:52) R and T (cid:52) S. We need to
show that T (cid:52) R∧ S = Mtdd. Observe that T (cid:52) M since T is semitransitive and M = R∧ S
T ST
is larger than any semitransitive relation smaller than both R and S. It implies in particular
that TInc ⊇MInc =(Mtdd)Inc and that TDec ⊆MDec.
Assume by contradiction that T (cid:54)(cid:52) Mtdd. Since TInc ⊇ (Mtdd)Inc, this means that there ex-
ist(b,a)∈TDec(cid:114)Mtdd. Wechoose(b,a)∈TDec(cid:114)Mtddsuchthatb−aisminimal. SinceTDec ⊆MDec,
we have (b,a) ∈ MDec (cid:114) Mtdd. By definition of Mtdd, there exists i ≤ b and j ≥ a such
that iMbMaMj while i(cid:54)Mj. Observe that bRa and bSa since (b,a)∈MDec =RDec∩SDec.
Since bMa, we cannot have both i = b and j = a. By symmetry, we can assume that i (cid:54)= b.
Since(i,b)∈MInc =(RInc∪SInc)tc,thereexistsi≤k <bsuchthat(i,k)∈MIncand(k,b)∈RInc∪SInc.
Assume without loss of generality that kRIncb. We obtain that kRbRa and thus that kRa by
transitivity of R. We now distinguish two cases:
• Assumethatk ≤a. Wethenhave(k,a)∈RInc ⊆MInc andthusthatiMInckMIncaMIncj
while i(cid:54)MIncj contradicting the transitivity of MInc.
• Assume that k ≥ a. Since (k,b) ∈ RInc ⊆ TInc and bTa we have k Ta by transitivity
of T. Since k ≥ a, we get (k,a) ∈ TDec ⊆ MDec. Therefore, we have i ≤ k and j ≥ a
with iMbMj and i(cid:54)Mj. This implies that (k,a) ∈ MDec(cid:114)Mtdd thus contradicting the
minimality of b−a. (cid:3)
Remark16. Wecanextractfromthepreviousproofthefollowingfactthatwillbeusedrepeatedly
inourproofs. LetRandSbetwotransitiverelations,letM=R∧ S,andlet1≤a<b≤nsuch
ST
that bMa and b(cid:54)Mtdda. By definition of Mtdd, there exist i≤b and j ≥a such that iMbMaMj
while i(cid:54)Mj. Then we have
• either i(cid:54)=b or j (cid:54)=a,
• if i(cid:54)=b, there is a<k <b such that iMkMb and (k,b)∈R∪S,
• if j (cid:54)=a, there is a<k <b such that aMkMj and (a,k)∈R∪S,
• in both cases, b(cid:54)Mtddk(cid:54)Mtdda.
Remark 17. In contrast to Propositions 6 and 9 and Corollaries 7 and 10, the cover relations
in(T(n),(cid:52))are morecomplicated todescribe. Infact, thelattice(T(n),(cid:52))is notgradedas soon
as n ≥ 3. Indeed, consider the maximal chains from I to D in (T(3),(cid:52)). Those chains passing
3 3
throughthetrivialreflexiverelation{(i,i)|i∈[n]}havealllength6, whilethosepassingthrough
the full relation [3]2 all have length 4.
1.2.3. Integer posets. We finally arrive to the subposet of the weak order induced by integer
posets. TheweakorderonP(3)isillustratedinFigure5. WenowhavealltoolstoshowTheorem1
announced in the introduction.
Proposition 18. The transitive meet ∧ and the transitive join ∨ both preserve antisymmetry.
T T
In other words, (P(n),(cid:52),∧ ,∨ ) is a sublattice of (T(n),(cid:52),∧ ,∨ ).
T T T T
Proof. Let R,S∈P(n). Let M:= R∧STS=(RInc∪SInc)tc∪(RDec∩SDec), so that R∧T S=Mtdd.
Assume that Mtdd is not antisymmetric. Let a < c ∈ [n] be such that {(a,c),(c,a)} ⊆ Mtdd with
c−a minimal. Since (c,a) ∈ (Mtdd)Dec ⊆ MDec = RDec ∩SDec, we have (a,c) ∈/ RInc ∪SInc by
antisymmetry of R and S. Since (a,c) ∈ (RInc ∪SInc)tc (cid:114)(RInc ∪SInc), there exists a < b < c
such that {(a,b),(b,c)}⊆(RInc∪SInc)tc. Since cMtdd aMtdd b, we obtain by transitivity of Mtdd
that {(b,c),(c,b)}⊆Mtdd, contradicting the minimality of c−a. (cid:3)
Remark 19. In contrast, there is no guarantee that the semitransitive meet of two transitive
antisymmetric relations is antisymmetric. For example in Figure 4, R and S are antisymmetric
butM=R∧ Sisnotasitcontainsboth(1,3)and(3,1). However,therelation(3,1)isremoved
ST
by the transitive decreasing delation and the result Mtdd =R∧ S is antisymmetric.
T
As in Propositions 6 and9 and Corollaries 7and 10, the next two statements describe allcover
relations in (P(n),(cid:52)).
Proposition 20. All cover relations in (P(n),(cid:52)) are cover relations in (R(n),(cid:52)). In particular,
(P(n),(cid:52)) is still graded by R(cid:55)→|RDec|−|RInc|.
8 GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
1 2 3
1 2 3 1 2 3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1 2 3
Figure 5. The weak order on integer posets of size 3.
Proof. Consider a cover relation R (cid:52) S in (P(n),(cid:52)). We have RInc ⊇ SInc and RDec ⊆ SDec
where at least one of the inclusions is strict. Suppose first that RInc (cid:54)= SInc. Consider the
set X:= (cid:8)(a,b)∈RInc(cid:114)SInc (cid:12)(cid:12)(cid:54)∃a<i<b with aRiRb(cid:9). This set X is nonempty as it contains
any (a,b) in RInc (cid:114)SInc with b−a minimal. Consider now (a,b) ∈ X with b−a maximal and
letT:=R(cid:114){(a,b)}. WeclaimthatTisstillaposet. Itisclearlystillreflexiveandantisymmetric.
Fortransitivity, assumebymeansofcontradictionthatthereisj ∈[n](cid:114){a,b}suchthataRjRb.
Since (a,b) ∈ X, we know that j < a or b < j. As these two options are symmetric, assume for
instance that j < a and choose j so that a−j is minimal. We claim that there is no j < i < b
such that jRiRb. Otherwise, since aRjRi and R is transitive, we have aRiRb. Now, if i<a,
wehaveaRiRbandj <i<acontradictingtheminimalityofa−j inourchoiceofj. Ifi>a,we
haveaRiRbanda<i<bcontradictingthefactthat(a,b)∈X. Finally,ifi=a,wehaveaRjRa
contradicting the antisymmetry of R. This proves that there is no j < i < b such that jRiRb.
By maximality of b−a in our choice of (a,b) this implies that jSb. Since (a,j)∈RDec ⊆SDec, we
therefore obtain that aSjSb while a(cid:54)Sb, contradicting the transitivity of S. This proves that T
is transitive and it is thus a poset. Moreover, we have R (cid:54)= T and R (cid:52) T (cid:52) S. Since S covers R,
this implies that S=T=R(cid:114){(a,b)}. We prove similarly that if RDec (cid:54)=SDec, there exists (b,a)
such that S=R∪{(b,a)}. In both cases, R(cid:52)S is a cover relation in (R(n),(cid:52)). (cid:3)
Corollary 21. In the weak order (P(n),(cid:52)), the posets that cover a given integer poset R∈P(n)
are precisely the posets
• the relations R(cid:114){(a,b)} for a<b such that aRb and there is no i∈[n] with aRiRb,
• the relations R∪{(b,a)} for a<b such that a(cid:54)Rb and b(cid:54)Ra and there is no i(cid:54)=a with aRi
but b(cid:54)Ri and similarly no j (cid:54)=b with jRb but j(cid:54)Ra.
THE WEAK ORDER ON INTEGER POSETS 9
Part 2. Weak order induced by some relevant families of posets
Intherestofthepaper,wepresentourmotivationtostudyTheorem1. Weobservethatmany
relevant combinatorial objects (for example permutations, binary trees, binary sequences, ...) can
be interpreted by specific integer posets. Moreover, the subposets of the weak order induced by
these specific integer posets often correspond to classical lattice structures on these combinatorial
objects (for example the classical weak order, the Tamari lattice, the boolean lattice, ...). Table 1
summarizes the different combinatorial objects involved and a roadmap to their properties.
As we will only work with posets, we prefer to use notations like (cid:67),(cid:74),(cid:97) which speak for
themselves, rather than our previous notations R,S,M for arbitrary binary relations. It also
allows us to write a (cid:66) b for b (cid:67) a, in particular when a < b. To make our presentation easier to
read, we have decomposed some of our proofs into technical but straightforward claims that are
proved separately in Appendix A.
2.1. From the permutahedron
We start with relevant families of posets corresponding to the elements, the intervals, and the
faces of the permutahedron. Further similar families of posets will appear in Sections 2.2 and 2.3.
Let S(n) denote the symmetric group on [n]. For σ ∈S(n), we denote by
ver(σ):=(cid:8)(a,b)∈[n]2 (cid:12)(cid:12)a≤b and σ−1(a)≤σ−1(b)(cid:9)
and inv(σ):=(cid:8)(b,a)∈[n]2 (cid:12)(cid:12)a≤b and σ−1(a)≥σ−1(b)(cid:9)
the set of versions and inversions of σ respectively1. Inversions are classical (although we order
their entries in a strange way), while versions are borrowed from [KLR03]. Clearly, the versions
of σ determine the inversions of σ and vice versa. The weak order on S(n) is defined as the
inclusion order of inversions, or as the clusion (reverse inclusion) order of the versions:
σ (cid:52)τ ⇐⇒ inv(σ)⊆inv(τ) ⇐⇒ ver(σ)⊇ver(τ).
Itisknownthattheweakorder(S(n),(cid:52))isalattice. Wedenoteby∧ and∨ itsmeetandjoin,
S S
and by e:=[1,2,...,n] and w◦:=[n,...,2,1] the weak order minimal and maximal permutations.
2.1.1. Weak Order Element Posets. We see a permutation σ ∈ S(n) as a total order (cid:67)
σ
on [n] defined by u (cid:67) v if σ−1(u) ≤ σ−1(v) (i.e. u is before v in σ). In other words, (cid:67) is the
σ σ
chain σ(1)(cid:67) ...(cid:67) σ(n) as illustrated in Figure 6.
σ σ
σ =2143 ←→ (cid:67)σ = 1 2 3 4
ver(σ)={(1,3),(1,4),(2,3),(2,4)} ←→ (cid:67)Iσnc = 1 2 3 4
inv(σ)={(2,1),(4,3)} ←→ (cid:67)Dσec = 1 2 3 4
Figure 6. A Weak Order Element Poset (WOEP).
We say that (cid:67) is a weak order element poset, and we denote by
σ
WOEP(n):=(cid:8)(cid:67)σ (cid:12)(cid:12)σ ∈S(n)(cid:9)
the set of all total orders on [n]. The following characterization of these elements is immediate.
Proposition22. Aposet(cid:67)∈P(n)isinWOEP(n)ifandonlyif∀u,v ∈[n],eitheru(cid:67)voru(cid:66)v.
The following proposition connects the weak order on S(n) to that on P(n). It justifies the
term “weak order” used in Definition 2.
1Throughoutthepaper, weonlyworkwithversionsandinversionsofvalues(sometimescalledleftinversions,
or coinversions). The cover relations of the weak order are thus given by transpositions of consecutive positions
(sometimescalledrightweakorder). Asthereisnoambiguityinthepaper,weneverspecifythisconvention.
10 GRE´GORYCHATEL,VINCENTPILAUD,ANDVIVIANEPONS
Permutreelattices permutrees[PP16] 742 3 5 641OPEP()Prop.60Thms.87&90dependsonOtheorientation Permutreelatticeintervals2121 3,3 11OPIP()Coro.55Coro.58dependsonOtheorientation Schr¨oderpermutrees[PP16] 742 3 5 641OPFP()Prop.63Rem.66dependsonOtheorientation
booleanlattice binarysequences−−−+++4356721 4356721 BOEPn()Prop.60Coro.88,,,,,...124816[OEIS,A000079] Booleanlatticeintervals321321 , 321321 BOIPn()Coro.55Coro.58,,,,,...1392781[OEIS,A000244] ternarysequences−−0+++7642135 3576421 BOFPn()Prop.63Rem.66,,,,,...1392781[OEIS,A000244] Part2.
n
ambrianlattices[Rea06] Cambriantrees[CP14] 763 5421 COEPε()Prop.60Coro.88,,,,,...1251442[OEIS,A000108] Cambrianlatticeintervals3232 , 11 COIPε()Coro.55Coro.58dependsonεthesignature Schr¨oderCambriantrees[CP14] 74235 61 COFPε()Prop.63Rem.66,,,,,...131145197[OEIS,A001003] orialobjectsconsideredi
C t
a
n
8 bi
Tamarilattice binarytrees 7235641 TOEPn()Prop.39rop.41&Coro.8,,,,,...1251442[OEIS,A000108] Tamarilatticeintervals , 123123 TOIPn()Coro.42Coro.44,,,,,...131368399[OEIS,A000260] Schr¨odertrees 3576421 TOFPn()Prop.46Rem.47,,,,,...131145197[OEIS,A001003] throughthecom
P p
a
m
weakorder permutations2751346 6435172 WOEPn()Prop.22Prop.24&Coro.91,,,,,...12624120[OEIS,A000142] weakorderintervals,[213321] 1213,32 WOIPn()Prop.26Coro.28,,,,,...13171511899[OEIS,A007767] orderedpartitions||1253746 6473 215 WOFPn()Prop.30Rem.31,,,,,...131375541[OEIS,A000670] Table1.Aroad
s s s
ne ne ne
oti oti oti
torial rizatiroper tion torial rizatiroper tion torial rizatiroper tion
a nep a a nep a a nep a
bincts tioactce mer bincts tioactce mer bincts tioactce mer
comobje notacharlatti enu comobje notacharlatti enu comobje notacharlatti enu
nts als
Eleme Interv Faces