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A NOTE ON OPERATORS OF DELETION AND CONTRACTION
FOR ANTICHAINS
ANDREY O. MATVEEV
Received 16 August 2001 and in revised form 28 March 2002
The operators of deletion and contraction for clutters are generalized to those for an-tichains of finite bounded posets. A generalization of the result by Seymour (1976), de-scribing the relationship between the operators of deletion, contraction, and the blocker map, is considered as a comparison in the lattice of antichains of a poset.
2000 Mathematics Subject Classification: 06A06, 90C27.
1. Introduction. Deletion and contraction are basic operators on clutters. Recall that for a finite nonempty setS, a family of its subsets is called aclutter(or aSperner family) if no set from that family contains another. Denote by ˆ0 the empty subset ofS. The clutter∅(containing no sets) and the clutter{ˆ0}are calledtrivial.
Consider a nontrivial clutterᏳon the ground setS. Letx∈S. Recall that thedeletion Ᏻ\xandcontractionᏳ/xare the clutters, defined as follows:
Ᏻ\x= {G∈Ᏻ:Gx},
Ᏻ/x=inclusion-wise minimal sets of the familyG−{x}:G∈Ᏻ, (1.1)
on the ground setS−{x}.
Deletion and contraction are also defined for the trivial clutters
∅\x= ∅/x= ∅, ˆ0\x=ˆ0/x=ˆ0, (1.2)
on the ground setS−{x}.
IfX= {x1,...,xn} ⊆S,n≥1, then thedeletionᏳ\XandcontractionᏳ/Xare defined in the following way:Ᏻ\X=Ᏻ\x1\···\xn, andᏳ/X=Ᏻ/x1/···/xn. Deletions and contractions, sequentially performed on a clutter, produce itsminors.
Theblocker of a nontrivial clutterᏳon the ground setS is the clutterᏮ(Ᏻ)onS, consisting of all the inclusion-wise minimal subsetsH⊆Swith the property, for each G∈Ᏻ,|H∩G| ≥1.
Theblockersof the trivial clutters are defined as follows (see, e.g., [2]):
Ꮾˆ0= ∅, Ꮾ(∅)=ˆ0. (1.3)
Seymour described in [4] the relationship between a clutterᏳon the ground setS, deletions, contractions, and relevant blockers; ifXis a nonempty subset ofSthen we have
Recall that the clutters on S are in natural one-to-one correspondence with the antichains of the Boolean lattice of all subsets ofS.
We present in this paper operators of deletion and contraction for antichains of a finite bounded poset. The main results of the paper are Theorems2.5and2.6.Theorem 2.5states that deletion and contraction for antichains are (co)closure operators on the lattice of antichains of the poset.Theorem 2.6provides a generalization of result (1.4) to antichains of the poset.
It is a consequence ofTheorem 2.6that equalities (1.4) might be read as
Ꮾ(Ᏻ)\X=Ꮾ(Ᏻ/X)Ꮾ(Ᏻ)Ꮾ(Ᏻ)/X=Ꮾ(Ᏻ\X), (1.5)
where is a certain comparison that comes from the lattice of antichains of the Boolean lattice of all subsets ofS.
2. Deletion and contraction. We refer the reader to [5, Chapter 3] for basic infor-mation and terminology in the theory of posets.
We use minQto denote the set of all minimal elements of a posetQ. IfQhas a least element then it is denoted ˆ0Q; ifQhas a greatest element then it is denoted ˆ1Q. Throughout this note,Pstands for a finite bounded poset with|P|>1;Padenotes its atom set, that is the set of all elements covering ˆ0P.I(A)andF(A)denote the order ideal and filter ofP generated by an antichainAofP, respectively.
We denote the distributive lattice of all antichains ofP byA(P). IfA1,A2∈A(P) then we set
A1≤A2 iffF
A1
⊆FA2
. (2.1)
The least and greatest elements ˆ0A(P)and ˆ1A(P)ofA(P)are thetrivial antichains∅ ⊂P and{ˆ0P}, respectively. For the rest of the paper, we denote by∧and∨the operations of meet and join in the latticeA(P); ifA1,A2∈A(P)then
A1∨A2=min
A1∪A2
, A1∧A2=min
FA1
∩FA2
, (2.2)
respectively.
LetAbe anontrivial antichainofP, that is A∈A(P)− {ˆ0A(P),ˆ1A(P)}. Theblocker
b(A)ofA, defined in [3], is the antichain
minb∈P:I(b)∩I(a)∩Pa≥1∀a∈A. (2.3)
Theblockersof the trivial antichains are defined as follows:
bˆ0A(P)=ˆ1A(P), bˆ1A(P)=ˆ0A(P). (2.4)
For a one-element antichain{a}ofP, we writeb(a)instead ofb({a}). Ifa≠ˆ0P then
b(a)=I(a)∩Pa, and we have{a} ≤b(b(a))≤b(a).
IfAis a nontrivial antichain ofPthen its blockerb(A)is determined, in particular, by the equalityb(A)=a∈Ab(a).
The mapb:A(P)→A(P), reflecting an antichain to its blocker, is theblocker map
A NOTE ON OPERATORS OF DELETION AND CONTRACTION... 727
imageb(A(P))is called in [3] thelattice of blockers inP and it is denotedB(P). The restrictionb|B(P)of the blocker map is an anti-automorphism ofB(P). The latticeB(P) is a meet-subsemilattice ofA(P). For every blockerB∈B(P), its preimageb−1(B)is a convex join-subsemilattice ofA(P); the greatest element ofb−1
(B)isb(B). We start with generalizing the notions of deletion and contraction.
Definition2.1. LetX⊆Pa,|X| ≥1.
(i) If{a}is a nontrivial one-element antichain ofP then thedeletion{a}\Xand
contraction{a}/Xare the antichains
{a}\X=
{a},
ifb(a)∩X=0, ˆ
0A(P), ifb(a)∩X≥1,
{a}/X=
{a}, ifb(a)∩X=0,
bb(a)−X, ifb(a)∩X≥1,b(a)X, ˆ
1A(P), ifb(a)⊆X.
(2.5)
(ii) IfAis a nontrivial antichain ofP then thedeletionA\X andcontractionA/X are the antichains
A\X= a∈A
{a}\X, A/X= a∈A
{a}/X. (2.6)
(iii) Thedeletionandcontractionfor the trivial antichains ofPare
ˆ
0A(P)\X=ˆ0A(P)/X=ˆ0A(P),
ˆ
1A(P)\X=ˆ1A(P)/X=ˆ1A(P).
(2.7)
If we take into consideration the empty subset∅aof the atom setPa, then for every A∈A(P)we define the antichainsA\∅aandA/∅ato be equal toA.
The following observations are an immediate consequence ofDefinition 2.1: if{a}
is a one-element antichain ofP, then we have
{a}\X≤ {a} ≤ {a}/X, (2.8)
b(a)\X≤b{a}/X≤b(a)≤b(a)/X=b{a}\X. (2.9)
Another observation is the following lemma.
Lemma2.2. LetX⊆Pa,|X| ≥1. IfA
1,A2∈A(P)andA1≤A2, then
A1\X≤A2\X, A1/X≤A2/X. (2.10)
Lemma2.3. IfA∈A(P)andX⊆Pa,|X| ≥1, then
A\X≤A≤A/X. (2.11)
Proof. There is nothing to prove ifAis a trivial antichain. Suppose thatAis non-trivial. With the help of (2.6) and (2.8), we see thatA\X=a∈A({a}\X)≤a∈A{a} = A≤a∈A({a}/X)=A/X.
If{a}is a nontrivial one-element antichain ofP andX⊆Pa,|X| ≥1, then we ob-viously have{a}\X=({a}\X)\X. The antichain{a}/Xhas an analogous property. Indeed, if|b(a)∩X| =0 or ifb(a)⊆Xthen({a}/X)/X= {a}/X, due to the defini-tion of contracdefini-tion. Further, if|b(a)∩X| ≥1 andb(a)X then, on one hand, we have({a}/X)/X≥ {a}/X, byLemma 2.3. On the other hand, for everyb∈ {a}/X=
b(b(a)−X)we haveb(b)−X≥b(a)−Xand, as a consequence, we have({a}/X)/X=
b∈{a}/X({b}/X)≤b(b(a)−X)= {a}/X. We conclude that({a}/X)/X= {a}/X. In view of (2.6), we can formulate the following lemma.
Lemma2.4. IfA∈A(P)andX⊆Pa,|X| ≥1, then
(A\X)\X=A\X, (A/X)/X=A/X. (2.12)
Altogether, Lemmas2.2, 2.3, and2.4describe the connection of the maps (\X):
A(P)→A(P)and(/X):A(P)→A(P)with (co)closure operators (see, e.g., [1, Chapter IV]).
Theorem2.5. LetX⊆Pa,|X| ≥1. The map(\X)is a coclosure operator onA(P). The map(/X)is a closure operator onA(P).
Given a nonempty atom subsetX, we denote, slightly abusing denotations, the im-ages(\X)(A(P))= {A\X:A∈A(P)}and(/X)(A(P))= {A/X:A∈A(P)}byA(P)\X and A(P)/X, respectively. We can interpret well-known properties of (semi)lattice maps and (co)closure operators on lattices in case of maps(\X)and(/X).
Definition 2.1 implies that the maps (\X),(/X):A(P)→A(P) are upper {ˆ0A(P), ˆ
1A(P)}-homomorphisms, that is for allA1,A2∈A(P), we have(A1∨A2)\X=(A1\X)∨ (A2\X),(A1∨A2)/X=(A1/X)∨(A2/X)and, moreover, we have ˆ0A(P)\X=ˆ0A(P)/X= ˆ
0A(P)and ˆ1A(P)\X=ˆ1A(P)/X=1ˆA(P).
The posetsA(P)\XandA(P)/X, with the partial orders induced by the partial order onA(P), are lattices.
The latticeA(P)\Xis a join-subsemilattice ofA(P). Denote by∧A(P)\Xthe operation of meet inA(P)\X. IfD1,D2∈A(P)\X, then we haveD1∧A(P)\XD2=(D1∧D2)\X.
The latticeA(P)/Xis a sublattice ofA(P).
IfD∈A(P)\X, then the preimage(\X)−1(D)ofDunder the map(\X)is the closed interval[D,D∨X]ofA(P).
IfD∈A(P)/X, then the preimage(/X)−1(D)ofDunder the map(/X)is a convex join-subsemilattice of the latticeA(P), with the greatest elementD.
A NOTE ON OPERATORS OF DELETION AND CONTRACTION... 729
the comparisons
b(A)\X≤b(A)≤b(A)/X, b(A/X)≤b(A)≤b(A\X) (2.13)
hold, and we make an additional conclusion in the following theorem.
Theorem2.6. IfA∈A(P)andX⊆Pa,|X| ≥1, then
b(A)\X≤b(A/X)≤b(A)≤b(A)/X≤b(A\X). (2.14)
Proof. There is nothing to prove if the antichainAis trivial.
IfA1,A2are arbitrary antichains ofPandX⊆Pa,|X| ≥1, then we can check with the help of routine machinery that
A1∧A2\X≤A1\X∧A2\X, (2.15)
A1∧A2
/X≤A1/X
∧A2/X
. (2.16)
Suppose thatAis nontrivial. We prove thatb(A)\X≤b(A/X). Using comparisons (2.15) and (2.9), we see that
b(A)\X=
a∈A
b(a)
\X≤ a∈A
b(a)\X≤ a∈A
b{a}/X=b
a∈A
{a}/X
=b(A/X).
(2.17) We prove thatb(A)/X≤b(A\X). With the help of (2.16) and (2.9), we see that
b(A)/X=
a∈A
b(a)
/X≤ a∈A
b(a)/X= a∈A
b{a}\X=b
a∈A
{a}\X
=b(A\X).
(2.18)
References
[1] M. Aigner,Combinatorial Theory, Grundlehren der Mathematischen Wissenschaften, vol. 234, Springer-Verlag, Berlin, 1979.
[2] R. Cordovil, K. Fukuda, and M. L. Moreira,Clutters and matroids, Discrete Math.89(1991), no. 2, 161–171.
[3] A. O. Matveev,On blockers in bounded posets, Int. J. Math. Math. Sci.26(2001), no. 10, 581–588.
[4] P. D. Seymour,The forbidden minors of binary clutters, J. London Math. Soc. (2)12(1976), no. 3, 356–360.
[5] R. P. Stanley,Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathe-matics, vol. 49, Cambridge University Press, Cambridge, 1997.
