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arX
iv:m
ath/
0010
092v
1 [
mat
h.C
O]
10
Oct
200
0
On blockers in bounded posets
Andrey O. Matveev
Data Center Company,
RU-620034, Toledova St. 43-A, Ekaterinburg, Russia
Abstract
Antichains of a bounded poset are assigned antichains playing arole analogous to that played by blockers in the Boolean lattice ofall subsets of a finite set. Some properties of lattices of generalizedblockers and maps on them are discussed.
1 Introduction
Blocking sets for finite families of finite sets, introduced by D.R. Fulkersonin the late 1960s, are important objects of discrete mathematics [1, 2].
A set H is called a blocking set for a family of sets G := {G1, . . . , Gm}if for each k ∈ {1, . . . ,m} we have H ∩Gk 6= ∅.
A family G is called a clutter if no set from it contains another.The blocker of G is the family H of all inclusion-minimal blocking sets
for G, it is a clutter.The following property [3] is basic. For a clutter G, the blocker of its
blocker coincides with G.The concepts of the blocker map and of the complementary map on
clutters made it possible to clarify relationship between families of specificsets and maps on them [4].
It is shown in this paper that the concepts of a blocking set and a blockercan be extended when passing from discussing clutters, considered antichainsof the Boolean lattice of all subsets of a finite set, to exploring antichains ofarbitrary bounded posets (a finite poset is called bounded if it has a uniqueminimal element, denoted 0, and a unique maximal element, denoted 1.)Some properties of the generalized blocker map and the complementarymap are discussed.
1
In Section 2 the notion of an intersecter plays a role analogous to thatplayed by blocking sets in the Boolean lattice of all subsets of a finite set.In Section 3 we explore the structure of sets of intersecters in Cartesianproducts of posets. In Section 4 and 6 properties of the blocker map andthe complementary map are discussed. In Section 5 we review the structureof lattices of blockers.
2 Intersecters and complementers
For a poset Q, we denote its atom set by Qa.For a subset X ⊆ Q, IQ(X) stands for the order ideal of Q generated
by X. For Q fixed, we also use the shorter notation I(X). The similarnotations FQ(X) or F(X) are used for the order ideal generated by X. Fora poset R, minR and maxR denote the sets of all minimal elements andall maximal elements of R respectively.
For a finite family of finite sets G, we denote its conventional blocker byM(G).
Throughout the paper P stands for a bounded poset such that P −{0, 1} 6= ∅.
:= denotes equality by definition.Let us start with extending of the concept of a blocking set.
Definition 1 Let A be a nonempty subset of P .
• If A 6= {0} then an element b ∈ P is an intersecter for A in P if forevery a ∈ A− {0} we have
I(b) ∩ I(a) ∩ P a 6= ∅ .
• For the set A := {0}, its unique intersecter in P is the minimal element0.
Every non-intersecter c for A is called a complementer for A in P .
Remark 1 Let B(n) denote the Boolean lattice of a finite rank n. If A is asubset of the poset B(n)−{0} then an element b ∈ B(n) is an intersecter forA if and only if I(b)∩B(n)a is a blocking set for the set family {I(a)∩B(n)a :a ∈ A}.
2
We use the notations I(P,A) and C(P,A) for the sets of all intersectersand all complementers for A in P respectively. For a one-element set {a},we use the shorter notation I(P, a) instead of I(P, {a}).
By definition, we have the partition
I(P,A)∪C(P,A) = P . (1)
For any subset A ⊆ P − {0}, the corresponding set of intersecters isnonempty because, at least, I(P,A) ∋ 1. In a similar way, ∅ 6= C(P,A) ∋ 0.
We view the sets I(P,A) and C(P,A) as the subposets of the poset P .It follows directly from the definition that for every subset A ⊆ P −{0},
I(P,A) = I(P,minA) , C(P,A) = C(P,minA) , (2)
therefore, in most cases, we can confine ourselves to studying intersectersfor antichains. Further,
I(P,A) =⋂
a∈A
I(P, a) , C(P,A) =⋃
a∈A
C(P, a) , (3)
and, finally, for all antichains A1, A2 6= {∅} of P such that F(A1) ⊇ F(A2),we have
I(P,A1) ⊆ I(P,A2) , C(P,A1) ⊇ C(P,A2) . (4)
Definition 1 implies the following reciprocity property for intersecters:for every antichain A of P ,
A ⊆ I(
P,min I(P,A))
, (5)
that is each element a ∈ A is an intersecter for the antichain min I(P,A) inP .
For an element a > 0, the corresponding set of intersecters I(P, a) is,obviously, the filter F
(
I(a) ∩ P a)
, so in accordance with (3), we obtain thefirst statement of the following lemma:
Lemma 1 Let A 6= {0} be an antichain of P . The set of intersecters for Ain P is determined by the following relations:
i.
I(P,A) =⋂
a∈A
F(
I(a) ∩ P a)
.
3
ii.
I(P,A) =⋃
E∈M({I(a)∩P a:a∈A})
(
⋂
e∈E
F(e))
.
Proof: To prove ii, note that the inclusion
I(P,A) ⊇⋃
E∈M({I(a)∩P a:a∈A})
(
⋂
e∈E
F(e))
follows from the definition of intersecters.We are left with proving the inclusion
I(P,A) ⊆⋃
E∈M({I(a)∩P a:a∈A})
(
⋂
e∈E
F(e))
.
Assume that it does not hold, and consider an intersecter b for A such thatb 6∈
⋃
E∈M({I(a)∩P a:a∈A})
(⋂
e∈E F(e))
. In this case, we have the inclusionb ∈
⋂
e∈E F(e) not for all atom subsets E from the family M({I(a) ∩ P a :a ∈ A}), hence there is an element a ∈ A such that I(b) ∩ I(a) ∩ P a = ∅.Therefore b is not an intersecter for A, but this contradicts to our choice ofb. Hence ii holds. �
Thus, the set of intersecters for any antichain A 6= {0} of the poset P is afilter of P , that is I(P,A) = F
(
min I(P,A))
. Consequently, the correspond-ing set of complementers is an ideal of P , that is C(P,A) = I
(
maxC(P,A))
.We call the elements of min I(P,A) the minimal intersecters or the blockers.We call the elements of maxC(P,A) the maximal complementers for A inP .
We denote the distributive lattice of all filters (partially ordered inclu-sionwise) of P by Filters(P ). If A 6= {0} is an antichain of P then, in termsof Filters(P ), Lemma 1 says:
Filters(P ) ∋ I(P,A) =∧
a∈A
∨
e∈I(a)∩P a
F(e)
=∨
E∈M({I(a)∩P a:a∈A})
∧
e∈E
F(e) ,(6)
where the filters F(e) are viewed as elements of Filters(P ), and joins andmeets are obtained in that lattice.
4
Proposition 1 Let P1 and P2 be bounded posets whose minimal elementsare denoted by 01 and 02 respectively. Let ψ : P1 → P2 be an order-preservingmap such that
ψ(01) = 02 ,
ψ(x1) > 02 for all x1 > 01 .(7)
For every antichain A1 of P1,
ψ(
I(P1, A1))
⊆ I(
P2, ψ(A1))
. (8)
Proof: If A1 = {01} then the proposition is true. Assume that A1 6= {01}.Let b1 be an intersecter for A1. In accordance with Definition 1, for alla1 ∈ A, we have P1
a ⊇ X1 := IP1(b1) ∩ IP1(a1) ∩ P1a 6= ∅. In conformity
with constraint (7), for every atom x1 ∈ X1, we have the following inclusion:
∅ 6= IP2
(
ψ(x1))
∩ P2a ⊆ IP2
(
ψ(a1))
∩ P2a .
Hence for all a2 ∈ ψ(A1), the inclusion b1 ∈ I(P,A1) implies
IP2
(
ψ(b1))
∩ IP2
(
ψ(a1))
∩ P2a 6= ∅ .
This means that ψ(b1) ∈ I(
P2, ψ(A1))
and completes the proof. �
3 Intersecters in Cartesian products of posets
In this section P1 and P2 stand for disjoint bounded posets with the minimalelements 01 and 02 and the maximal elements 11, 12. We assume that P1 :=P1 − {01, 11} 6= ∅ and P2 := P2 − {02, 12} 6= ∅.
Proposition 2 We denote the Cartesian product of the posets P1 and P2
by P1 ×P2. Let P := (P1 ×P2)∪{0, 1}, where 0, 1 are the new minimal andmaximal elements. Let A be an antichain of P := P1 × P2. We define
A ⇂P1 := {a1 ∈ P1 : (a1; a2) ∈ A} and A ⇂P2:= {a2 ∈ P2 : (a1; a2) ∈ A} .
i. If min I(P1, A ⇂P1) = {11} or min I(P2, A ⇂P2) = {12} then
I(P,A) = min I(P,A) = {1} .
5
ii. If min I(P1, A ⇂P1) 6= {11} and min I(P2, A ⇂P2) 6= {12} then
I(P,A) =(
I(P1, A ⇂P1) − {11})
×(
I(P2, A ⇂P2) − {12})
∪ {1}
and
min I(P,A) = min I(P1, A ⇂P1) × min I(P2, A ⇂P2) .
Proof: The atom set P a is P1a × P2
a, therefore, by Lemma 1 i, the set ofintersecters has the form:
I(P,A) =⋂
a:=(a1;a2)∈A
FP
(
(
IP1(a1) × IP2(a2))
∩(
P1a × P2
a)
)
∪ {1}
=
⋂
a1∈A⇂P1
FP1
(
IP1(a1) ∩ P1a)
×
⋂
a2∈A⇂P2
FP2
(
IP2(a2) ∩ P2a)
∪ {1}
=(
I(P1, A ⇂P1) − {11})
×(
I(P2, A ⇂P2) − {12})
∪ {1} ,
from where the statement follows. �
Now we shall study the structure of the set of intersecters in a slightlydifferent construction.
Lemma 2 Let P be the Cartesian product P := P1 × P2. Let A be anantichain of P := P − {0, 1}. Then
I(P,A) =
(
P1 ×(
⋂
(a1;a2)∈A
FP2
(
IP2(a2) ∩ P2a)
)
)
∪
(
(
⋂
(a1;a2)∈A
FP1
(
IP1(a1) ∩ P1a)
)
× P2
)
.
(9)
Proof: Since for every element x := (x1;x2) ∈ P it holds
IP1×P2(x) = IP1(x1) × IP2(x2) ,
and the atom set of the poset P has the form
P a =(
{01} × P2a)
∪(
P1a × {02}
)
,
6
we can, using Lemma 1 i, write down
I(P,A) =⋂
(a1;a2)∈A
FP
(
(
IP1(a1) × IP2(a2))
∩(
(
{01} × P2a)
∪(
P1a × {02}
)
)
)
,
from where (9) follows. �
Proposition 3 Let P conform to the hypothesis of Lemma 2. If A is anantichain of P := P − {0, 1} such that
{(a1; 02) ∈ A : a1 > 01} 6= ∅ ,
{(01; a2) ∈ A : a2 > 02} 6= ∅ ,
{(a1; a2) ∈ A : a1 > 01, a2 > 02} 6= ∅ ,
then
I(P,A) =(
I(
P1, {a1 > 01 : (a1; 02) ∈ A})
× I(P2, A ⇂P2))
∪(
I(P1, A ⇂P1) × I(
P2, {a2 > 02 : (01; a2) ∈ A})
) (10)
and
min I(P,A) =(
min I(
P1, {a1 > 01 : (a1; 02) ∈ A})
× min I(P2, A ⇂P2))
∪(
min I(P1, A ⇂P1) × min I(
P2, {a2 > 02 : (01; a2) ∈ A})
)
.
(11)
Proof: Using Lemma 2, we rewrite (9):
I(P,A) =
(
⋂
(a1;02)∈A:
a1>01
FP1
(
IP1(a1) ∩ P1a)
)
× P2
)
∩
(
P1 ×(
⋂
(01;a2)∈A:
a2>02
FP2
(
IP2(a2) ∩ P2a)
)
∩
(
P1 ×(
⋂
(a1;a2)∈A:
a1>01,a2>02
FP2
(
IP2(a2) ∩ P2a)
)
)
∪
(
(
⋂
(a1;a2)∈A:
a1>01,a2>02
FP1
(
IP1(a1) ∩ P1a)
)
× P2
)
.
7
After processing we obtain
I(P,A) =
(
⋂
(a1;02)∈A:
a1>01
FP1
(
IP1(a1) ∩ P1a)
)
×(
⋂
a2∈A⇂P2
FP2
(
IP2(a2) ∩ P2a)
)
∪
(
⋂
a1∈A⇂P1
FP1
(
IP1(a1) ∩ P1a)
)
×(
⋂
(01;a2)∈A:
a2>02
FP2
(
IP2(a2) ∩ P2a)
)
,
from where (10) follows. (11) is its corollary. �
Proposition 4 Let P conform to the hypothesis of Lemma 2. If A is anantichain of P such that
A ⇂P1 6∋ 01 and A ⇂P2 6∋ 02 ,
then
I(P,A) =(
P1 × I(P2, A ⇂P2))
∪(
I(P1, A ⇂P1) × P2
)
(12)
and
min I(P,A) =(
{01} × min I(P2, A ⇂P2))
∪(
min I(P1, A ⇂P1) × {02})
.
(13)
Proof: (9) is rewritten here in the following way:
I(P,A) =
P1 ×(
⋂
a2∈A⇂P2
FP2
(
IP2(a2) ∩ P2a)
)
∪
(
⋂
a1∈A⇂P1
FP1
(
IP1(a1) ∩ P1a)
)
× P2
.
(12) is equivalent to it, and (13) follows directly. �
Proposition 5 Let A be an antichain of the poset P := (P1 × P2)−{0, 1}.
8
i. If
A ⇂P1= {01} and A ⇂P2 6= {02} ,
then
I(P,A) = P1 × I(
P2, {a2 > 02 : (01; a2) ∈ A})
(14)
and
min I(P,A) = {01} × min I(
P2, {a2 > 02 : (01; a2) ∈ A})
. (15)
ii. If
A ⇂P1 6= {01} and A ⇂P2= {02} ,
then
I(P,A) = I(
P1, {a1 > 01 : (a1; 02) ∈ A})
× P2
and
min I(P,A) = min I(
P1, {a1 > 01 : (a1; 02) ∈ A})
× {02} .
Proof: Under the hypothesis of i, (9) gets the form:
I(P,A) = P1 ×(
⋂
(01;a2)∈A:
a2>02
FP2
(
IP2(a2) ∩ P2a)
)
,
that is equivalent to (14), from where (15) follows. ii is analogous to i inaccordance with commutativity of Cartesian products. �
Proposition 6 Let A be an antichain of the poset P := (P1 × P2)−{0, 1}.
i. If
{(a1; 02) ∈ A : a1 > 01} 6= ∅ ,
{(01; a2) ∈ A : a2 > 02} = ∅ ,
{(a1; a2) ∈ A : a1 > 01, a2 > 02} 6= ∅ ,
9
then
I(P,A) =(
I(
P1, {a1 > 01 : (a1; 02) ∈ A})
× I(
P2, {a2 > 02 : (a1; a2) ∈ A, a1 > 01})
)
∪(
I(
P1, {a1 > 01 : (a1; a2) ∈ A})
× P2
)
(16)
and
min I(P,A) =
(
(
min I(
P1, {a1 > 01 : (a1; 02) ∈ A})
− min I(
P1, A ⇂P1
)
)
× min I(
P2, {a2 > 02 : (a1; a2) ∈ A, a1 > 01})
)
∪
(
min I(
P1, A ⇂P1
)
× {02}
)
.
(17)
ii. If
{(a1; 02) ∈ A : a1 > 01} = ∅ ,
{(01; a2) ∈ A : a2 > 02} 6= ∅ ,
{(a1; a2) ∈ A : a1 > 01, a2 > 02} 6= ∅ ,
then
I(P,A) =(
I(
P1, {a1 > 01 : (a1; a2) ∈ A, a2 > 02})
× I(
P2, {a2 > 02 : (01; a2) ∈ A})
)
∪(
P1 × I(
P2, {a2 > 02 : (a1; a2) ∈ A})
)
and
min I(P,A) =
(
min I(
P1, {a1 > 01 : (a1; a2) ∈ A, a2 > 02})
×(
min I(
P2, {a2 > 02 : (01; a2) ∈ A})
− min I (P2, A ⇂P2))
)
∪
(
{01} × min I (P2A ⇂P2)
)
.
10
Proof: In conformity with the hypothesis of i, (9) is rewritten in the fol-lowing way:
I(P,A) =
(
⋂
(a1;02)∈A:
a1>01
FP1
(
IP1(a1) ∩ P1a)
)
× P2
∩
(
P1 ×(
⋂
(a1;a2)∈A:
a1>01,a2>02
FP2
(
IP2(a2) ∩ P2a)
)
)
∪
(
(
⋂
(a1;a2)∈A:
a1>01,a2>02
FP1
(
IP1(a1) ∩ P1a)
)
× P2
)
.
After processing we obtain
I(P,A) =
(
⋂
(a1;02)∈A:
a1>01
FP1
(
IP1(a1) ∩ P1a)
)
×(
⋂
(a1;a2)∈A:
a1>01,a2>02
FP2
(
IP2(a2) ∩ P2a)
)
∪
(
⋂
(a1;a2)∈A:
a1>01
FP1
(
IP1(a1) ∩ P1a)
)
× P2
,
that is equivalent to (16). (17) is its corollary. ii is equivalent to i, in accor-dance with commutativity of Cartesian products. �
4 Blocker map and complementary map
We denote the set of all (nonempty) antichains of P by Ant(P ). We meanthat the set Ant(P ) is partially ordered: for all antichains A′ and A′′, we set
A′ ≤Ant(P ) A′′ if and only if FP (A′) ⊆ FP (A′′) ;
in other words, we use the order isomorphism(
Filters(P )−{0})
→ Ant(P )such that F 7→ minF .
11
Recall (cf., for instance, with [5]) that for A′, A′′ ∈ Ant(P ),
A′ ∨Ant(P ) A′′ = min(A′ ∪A′′) and A′ ∧Ant(P ) A
′′ = min(
FP (A′) ∩ FP (A′′))
.
Let b : Ant(P ) → Ant(P ) be the blocker map: by definition,
b : A 7→ min I(P,A) ;
in particular, for a ∈ P such that a > 0, we have b({a}) = I(a) ∩ P a.For a one-element antichain {a}, we write b(a) instead of b({a}).Let Antb(P ) := im(b) denote the image of the map b. The set Antb(P )
is equipped, by definition, with the partial order induced by the partial orderon Ant(P ).
For each antichain B of P , we define the preimage of B under the mapb, in the following way:
b−1(B) := {A ∈ Ant(P ) : b(A) = B} .
We have Ant(P ) = ˙⋃B∈Antb(P ) b−1(B), here b−1({0}) = {0}.
We have, at least, the following inclusions:
• Antb(P ) ∋ {0}.
• Antb(P ) ⊃{
b(a) : a ∈ P − {0}}
; in particular, Antb(P ) ∋ P a = b(1).
• Antb(P ) ⊃{
min⋂
y∈b(a) F(y) : a ∈ P − {0}}
.
Reformulate (4):
Lemma 3 If A′ ≤Ant(P ) A′′ <Ant(P ) 1 then b(A′) ≥Ant(P ) b(A′′).
In the theory of blocking sets the following fact is basic:
Proposition 7 [3] For any clutter G, we have M(
M(G))
= G.
This proposition can be generalized in the following way:
Theorem 1 The restriction map b |Antb(P ) is an involution, that is for eachantichain B ∈ Antb(P ) we have
b(
b(B))
= B .
12
Proof: The theorem is evidently true when B = {0}.Consider the situation when B 6= {0}. Choose an antichain A′ ∈ b−1(B).
In accordance with the reciprocity property for intersecters, every elementof A′ is an intersecter for the antichain B = b(A′). In other words, for eachelement a′ ∈ A′, we have the inclusion a′ ∈ I(P,B) =
⋂
b∈B F(
I(b)∩P a)
. Inaccordance with this inclusion, we assign to the antichain A′ the antichain
A := min⋂
b∈B
F(
I(b) ∩ P a)
∈ b−1(B)
(that may coincide with A′), which is a blocker for B, by Lemma 1 i. Sothen b(A) = B, b(B) = A, and the theorem follows. �
Example 1 Figure 1 (b) shows the Hasse diagram of the poset Antb(P ) forthe poset P with the diagram shown in Figure 1 (a). We have
Antb(P ) ={
{0}{x}{z}{w}{y}{v}{1}{xz}{zw}{xzw}}
,
see Figure 1 (c). Here
b−1({0}) ={
{0}}
, b−1({x}) ={
{x}}
, b−1({z}) ={
{z}{yv}}
,
b−1({w}) ={
{w}}
, b−1({y}) ={
{xz}{xv}}
, b−1({v}) ={
{zw}{yw}}
,
b−1({1}) ={
{xw}{xzw}}
, b−1({xz}) ={
{y}}
, b−1({zw}) ={
{v}}
,
b−1({xzw}) ={
{1}}
.
The restriction map b |Antb(P ):
{0} 7→ {0}, {x} 7→ {x}, {z} 7→ {z}, {w} 7→ {w}, {xz} 7→ {y},
{y} 7→ {xz}, {zw} 7→ {v}, {v} 7→ {zw}, {1} 7→ {xzw}, {xzw} 7→ {1}.
The set{
{0}}
∪{
{x} : x ∈ P a}
is, evidently, the set of fixed points ofthe restriction map b |Antb(P ). But this map can have other fixed points asit can be seen from the following:
Example 2 For the Boolean lattice B(3) of rank 3, the set B(3)(2) of allelements of the rank 2 is a fixed point of the map b |Antb(B(3)).
In general, by Lemma 1, an antichain A 6= {0} of P is a fixed point ofb |Antb(P ) if and only if the following equivalent relations hold:
13
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{y}
{x}
{xz}
{v}
{w}
{zw}
{z}
{xzw}
{1 ∈ P}
{0 ∈ P}
(c)
Antb(P )
Figure 1:
14
•
FP (A) =⋂
a∈A
FP(
b(a))
. (18)
•
FP (A) =⋃
E∈M({b(a):a∈A})
(
⋂
e∈E
FP (e)
)
.
Let {0 ∈ P} 6= B ∈ Antb(P ) and A′, A′′ ∈ b−1(B). By Lemma 1 i,
B = min⋂
a′∈A′
FP(
b(a′))
= min⋂
a′′∈A′′
FP(
b(a′′))
.
Therefore
B = min
(
(
⋂
a′∈A′
FP(
b(a′))
)
∩(
⋂
a′′∈A′′
FP(
b(a′′))
)
)
= min⋂
a∈min(A′∪A′′)
FP(
b(a))
= min⋂
a∈A′∨Ant(P )A′′
FP(
b(a))
.
Hence A′ ∨Ant(P ) A′′ ∈ b−1(B).
Note that if B = {0} then {0} ∨Ant(P ) {0} = {0} = b−1({0}).We have proven
Proposition 8 For each antichain B ∈ Antb(P ), the preimage b−1(B) is ajoin-subsemilattice of the distributive lattice Ant(P ). In particular, b−1(B)has the maximal element b(B).
Remark 2 For A′, A′′ ∈ b−1(B), the inclusion A′ ∧Ant(P ) A′′ ∈ b−1(B) does
not hold, in general, as Figure 2 shows.
Let c : Ant(P ) → Ant(P ) be the complementary map, that is, by defini-tion,
c : A 7→ maxC(P,A) .
We use the notation Antc(P ) := im(c) for the image of the map c. Theset Antc(P ) is equipped, by definition, with the partial order induced by the
15
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0
1P
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@r r r
r
{xy} {xz} {yz}
{xyz}
b−1({vwp})
Figure 2:
partial order on the lattice of ideals of P : for all C ′, C ′′ ∈ Antc(P ), we setC ′ ≤ C ′′ if and only if IP (C ′) ⊆ IP (C ′′).
For each antichain C of P , we define the preimage of C under the mapc in the following way:
c−1(C) := {A ∈ Ant(P ) : c(A) = C} ,
hence Ant(P ) = ˙⋃C∈Antc(P ) c−1(C).
The poset Antc(P ) includes, at least, the set of antichains {max(P −F(
I(a)∩P a))
: 0 < a ∈ P} that correspond to one-element antichains of P .
Example 3 For the poset P with the diagram shown in Figure 3 (comparewith Figure 5),
Antc(P ) ={
{0}{z}{v}{1}{yz}{xz}{zv}{xyz}}
.
Here
c−1({0}) ={
{1}}
, c−1({z}) ={
{v}}
, c−1({yz}) ={
{x}}
,
c−1({xz}) ={
{y}}
, c−1({v}) ={
{z}}
, c−1({xyz}) ={
{xy}}
,
c−1({zv}) ={
{xz}{yz}{zv}{xyz}}
, c−1({1}) ={
{0}}
.
Proposition 9 An antichain A ∈ Ant(P ) is a fixed point of the comple-mentary map c if and only if we have the following partition:
FP(
b(A))
∪IP (A) = P . (19)
Proof: For the antichain {0} (which is not a fixed point of c), (19) does nothold.
16
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@@
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@
P
r
r
r
r
r
rx y
v
z
0
1
@@
@
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�
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@@
��
�
@@
@r
r
r
r
r
r
r
r{yz}
{z}
{xyz}
{xz}
{zv}
{v}
{0 ∈ P}
{1 ∈ P}
Antc(P )
Figure 3:
Consider an antichain A 6= {0}. According to (18), A is a fixed point of
c if and only if we have the partition max(
P − FP(
c(A))
)
= A or, equiva-
lently, P − FP(
b(A))
= IP (A), from where the statement follows. �
5 Lattice of blockers
Lemma 1 i implies that for all B′, B′′ ∈ Antb(P ) fixed, the set b(A′ ∧Ant(P )
A′′) is the same antichain for all antichains A′ ∈ b−1(B′) and A′′ ∈ b−1(B′′),namely
b(A′ ∨Ant(P ) A′′) = B′ ∧Ant(P ) B
′′ . (20)
Proposition 10 For all B′, B′′ ∈ Antb(P ),
B′ ∧Ant(P ) B′′ = b
(
b(B′) ∨Ant(P ) b(B′′))
.
Proof: The proposition is evidently true when B′ = B′′ = {0}.Let B′, B′′ 6= {0}. Using Theorem 1, we write down
B′ ∧Ant(P ) B′′ = b
(
b(B′))
∧Ant(P ) b(
b(B′′))
.
17
Continue with processing:
B′ ∧Ant(P ) B′′ = min
(
F
(
b(
b(B′))
)
∩ F
(
b(
b(B′′))
)
)
= min(
⋂
a′∈b(B′)
F(
b(a′))
∩⋂
a′′∈b(B′′)
F(
b(a′′))
)
= min⋂
a∈b(B′)∪b(B′′)
F(
b(a))
= min⋂
a∈min
(
b(B′)∪b(B′′))
F(
b(a))
= b
(
min(
b(B′) ∪ b(B′′))
)
= b(
b(B′) ∨Ant(P ) b(B′′))
.
The proposition is also true when B′ = {0}, B′′ 6= {0}. �
Remark 3 So for all B′, B′′ ∈ Antb(P ), we have the inclusion B′ ∧Ant(P )
B′′ ∈ Antb(P ). Hence Antb(P ) is a meet-subsemilattice of the lattice Ant(P );the minimal element of Antb(P ) is b(P a).
Remark 4 In general, Antb(P ) is not a join-subsemilattice of the latticeAnt(P ), as it can be seen from the following:
Example 4 For the poset P with the diagram shown in Figure 4, {x}∨Ant(P )
{z} = {xz}, but {x} ∨Antb(P ) {z} = {xyz}.
Remark 5 In general, for A′, A′′ ∈ Ant(P ) relation b(A′) ∨Antb(P ) b(A′′) =b(A′ ∧Ant(P ) A
′′) does not hold, as it can be seen from the following:
Example 5 See Figure 4; here
b({xy}) ∨Antb(P ) b(z) = {vz} 6= b({xy} ∧Ant(P ) {z}) = {yz} .
One can deduce from the reciprocity property for intersecters the follow-ing lemma:
18
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r
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r
r
r
r
r
{vw}
{x}
{xy}
{wp}
{z}
{yz}
{y}
{xyz}
{w}
{0}
Antb(P )
Figure 4:
Lemma 4
i. If we have B′ ≤ B′′ for B′, B′′ ∈ Antb(P ) then B′ is a set of cer-tain intersecters for the antichain b(B′′) and, consequently, for eachmember of b−1(B′′).
ii. If we have C ′ ≤ C ′′ for C ′, C ′′ ∈ Antc(P ) then C ′ is a set of certain
complementers for the antichain b
(
min(
P−IP (C ′′))
)
∈ c−1(C ′′) and,
consequently, for each member of c−1(C ′′).
Corollary 1
i. The poset Antb(P ) − {1} is self-dual.
ii. For each B ∈ Antb(P ), the elements B and b(B) are comparable inAntb(P ).
Proof: The restriction map b |Antb(P )−{1} is an order antiisomorphism of
the posets Antb(P ) − {1} with B 7→ b(B) and b(B) 7→ B. �
Theorem 2 The poset Antb(P ) is a (not distributive, in general) latticewith the minimal element b(P a), with the maximal element {0 ∈ P} andwith the unique coatom P a. Moreover
i. Antb(P ) is a meet-subsemilattice (not a join-subsemilattice, in generalcase) of the lattice Ant(P ).
19
ii. The lattice Antb(P ) − {1} is self-dual.
iii. In the lattice Antb(P ) − {1} operations of obtaining meets and joinsare determined as follows: for B′, B′′ ∈ Antb(P ) − {1}
B′ ∧Antb(P ) B′′ = B′ ∧Ant(P ) B
′′ (21)
and
B′ ∨Antb(P ) B′′ = b(B′ ∧Ant(P ) B
′′) . (22)
Proof: We are left with proving iii; (21) has been proven above, see Re-mark 3.
By self-duality of the lattice Antb(P )−{1}, B′∧Antb(P )B′′ = B′∧Ant(P )
B′′ = b(B′ ∨Antb(P )B′′). Using Theorem 1, we obtain (22): B′∧Ant(P )B
′′ =b(
b(B′ ∨Antb(P ) B′′))
= b(B′ ∧Antb(P ) B′′). �
Remark 6 In general, for antichains B′, B′′ ∈ Antb(P ) fixed, the antichainb(A′∧Ant(P )A
′′) is not the same set for different antichains A′ ∈ b−1(B′), A′′ ∈
b−1(B′′) as it can be seen from the following:
Example 6 Look at Figure 5. Let B′ := {v} and B′′ := {1}. Here b−1(B′) ={
{xy}}
and b−1(B′′) ={
{xz}{yz}{xyz}{zv}}
. We have the following in-clusions:
{xy} ∧Ant(P ) {xz} = {x} ∈ b−1({x}) ,
{xy} ∧Ant(P ) {yz} = {y} ∈ b−1({y}) ,
{xy} ∧Ant(P ) {xyz} = {xy} ∈ b−1({v}) ,
{xy} ∧Ant(P ) {zv} = {v} ∈ b−1({xy}) .
It follows directly from the definition of the complementary map that:
• The restriction
c |Antb(P ): Antb(P ) → Antc(P )
of the complementary map, under which
c |Antb(P ): B 7→ c(B) ,
is an order isomorphism between the lattices Antb(P ) and Antc(P ).
20
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P
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rx y
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@@
@r
r
r
r
r
r
r
r{x}
{v}
{xy}
{y}
{xyz}
{z}
{1 ∈ P}
{0 ∈ P}
Antb(P )
Figure 5:
• The reverse order isomorphism
(c |Antb(P ))−1 : Antc(P ) → Antb(P )
is the map, under which
(c |Antb(P ))−1 : C 7→ b
(
min(
P − IP (C))
)
.
• For every element C ∈ Antc(P ), we have | c−1(C) ∩ Antb(P )| = 1,namely,
c−1(C) ∩ Antb(P ) = b
(
min(
P − IP (C))
)
.
We define the subposet
Anta(P ) := {X ∈ Antb(P ) : X ⊆ P a}
of the poset Antb(P ). Hence Anta(P ) ⊆ Antb(P ) ⊆ Ant(P ) and, denotingthe Boolean lattice of all subsets of the atom set P a by B(P a), we haveAnta(P ) = Antb(P ) ∩
(
B(P a) − {0})
and minAnta(P ) ={
{x} : x ∈ P a}
.Using Lemma 1 i and properties of the lattice Antb(P ), we obtain the
following:
Proposition 11 Each of the posets Anta(P ) and {b(X) : X ∈ Anta(P )} isan irredundant set of generators for the lattice Antb(P ) − {1}.
Corollary 2 Bounded posets P and Q have the order isomorphic posetsAntb(P ) and Antb(Q) if and only if the posets Anta(P ) and Anta(Q) areorder isomorphic.
21
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(a)
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r
Anta(·) Antb(·)
(b)
Figure 6:
Example 7 Four posets having the diagrams shown in Figure 6 (a) havethe order isomorphic posets Anta(·) and, consequently, they have the orderisomorphic posets Antb(·); their diagrams are shown in Figure 6 (b).
Let S be such a meet-subsemilattice of the Boolean lattice B(n) of afinite rank n that
• S has the maximal element that coincides with the maximal elementof B(n),
• the minimal element of S coincides with the minimal element 0 ofB(n),
• the atom set of S coincides with the atom set of B(n).
It is easy to see, using Proposition 10, that the posets S −{0} and Anta(S)are order isomorphic. Hence Proposition 11 implies that for a poset P thereexists such a meet-subsemilattice S of a Boolean lattice that the latticesAntb(P ) and Antb(S) are order isomorphic.
22
6 Order-preserving maps and blockers
If ψ is an endomorphism of a poset P then, in accordance with Lemma 3,
b(
minψ(A))
≥Antb(P ) b(A) ,
because minψ(A) ≤Ant(P ) A.Let ψ be an endomorphism of P such that
ψ(0) = 0 ; ψ(a) > 0 for a > 0 . (23)
If A 6= {0} is an antichain of P then, in conformity with relation (2) andLemma 1 i,
b(
minψ(A))
= b(
ψ(A))
= min⋂
a∈A
FP
(
b(
ψ(a))
)
= min
(
min⋂
a∈A
FP
(
b(
ψ(a))
∩ b(a))
∪ min⋂
a∈A
FP
(
b(
ψ(a))
⊖ b(a))
)
,
where ⊖ denotes the symmetric difference of sets.Since b
(
ψ(a))
⊇ b(a) for each a ∈ A, we obtain
b(
minψ(A))
= min
(
min⋂
a∈A
FP(
b(a))
∪ min⋂
a∈A
FP
(
b(
ψ(a))
− b(a))
)
.
Using the notation
Dψ(A) := min⋂
a∈A
FP
(
b(
ψ(a))
− b(a))
,
we rewrite it in an equivalent way:
b(
minψ(A))
= min(
b(A) ∪Dψ(A))
,
We have obtained
Proposition 12 Let ψ be an endomorphism of P such that condition (23)holds. If A 6= {0} is an antichain of P then
b(
minψ(A))
= b(A) ∨Ant(P ) Dψ(A) . (24)
23
Remark 7
• Under the hypothesis of Proposition 12
b
(
b(
minψ(A))
)
= b(
b(A) ∨Ant(P ) Dψ(A))
= b(
b(A))
∧Ant(P ) b(
Dψ(A))
= b(
b(A))
∧Antb(P ) b(
Dψ(A))
.
(25)
• If {0 ∈ P} 6= R ∈ Antb(P ) then (25) implies, in accordance withTheorem 1,
b
(
b(
minψ(R))
)
= R ∧Antb(P ) b(
Dψ(R))
.
Furthermore, if ψ(R) ∈ Antb(P ) then (25) implies
ψ(R) = R ∧Antb(P ) b(
Dψ(A))
.
Let {p1, p2} be a two-element antichain of P and ψ an endomorphism ofP satisfying condition (23). By Proposition 12,
b(
ψ(p1))
= b(p1) ∨Ant(P ) Dψ({p1}) ,
b(
ψ(p2))
= b(p2) ∨Ant(P ) Dψ({p2}) .
Therefore, see (20),
b(
minψ({p1, p2}))
= b(
ψ(p1))
∧ b(
ψ(p2))
=(
b(p1) ∧ b(p2))
∨(
b(p1) ∧ Dψ(p2))
∨(
b(p2) ∧ Dψ(p1))
∨(
Dψ(p1) ∧Dψ(p2))
(operations ∨ and ∧ are realized in Ant(P )), note that here b(p1) ∧Ant(P )
b(p2) = b({p1, p2}).In general, discoursing by induction, we obtain: for an antichain A 6= {0}
of P with |A| > 1,
Dψ(A) =(
∧
a∈A
Dψ(a))
∨∨
1≤i≤|A|−1
(
∨
A′⊂A:|A′|=i
(
b(A′) ∧∧
a∈A−A′
Dψ(a))
)
(all operations are realized in Ant(P )).
24
Let P and Q be bounded posets. We denote the poset of all the order-preserving maps from P into Q by QP . By definition, for ψ′, ψ′′ ∈ QP ,ordering ψ′ ≤ ψ′′ holds if and only if ψ′(p) ≤ ψ′′(p) for all p ∈ P .
Let ψ′ ∈ QP be a map such that
ψ′(0 ∈ P ) = 0 ∈ Q; ψ′(p) > 0 ∈ Q for all p ∈ P such that p > 0 .
For an antichain A 6= {0} of P , we use the notation
Dψ′≤ψ′′(A) := min⋂
a∈A
FP
(
b(
ψ′′(a))
− b(
ψ′(a))
)
. (26)
The following relation holds:
b(
minψ′′(A))
= b(
minψ′(A))
∨Ant(Q) Dψ′≤ψ′′(A) . (27)
(27) generalizes formula (24) for endomorphisms. In notations (26), rela-tion (27) is written as follows:
b(
minψ(A))
= b(
Id(A))
∨Ant(P ) DId≤ψ(A) ,
where Id denotes the identity map on P .
References
[1] M. Grotschel, L. Lovasz and A. Schrijver, Geometric algorithms andcombinatorial optimization (Springer, Berlin Heidelberg New York,1993).
[2] Z. Furedi, Matchings and covers in hypergraphs, Graphs and combina-torics. 4 (1988) 115–206.
[3] J. Edmonds and D.R. Fulkerson, Bottleneck extrema, Journal of combi-natorial theory. 8 (1970) 299–306.
[4] R. Cordovil, K. Fukuda and M.L. Moreira, Clutters and matroids, Dis-crete mathematics. 89 (1991) 161–171.
[5] C. Greene and D.J. Kleitman, The structure of Sperner k-families, Jour-nal of combinatorial theory. Ser. A 20 (1976) 41–68.
25
