On blockers in bounded posets

Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.


Introduction
Blocking sets for finite families of finite sets are important objects of discrete mathematics (see [5, Chapter 8], [3]).
A set H is called a blocking set for a nonempty family G = {G 1 , . . . , G m } of nonempty subsets of a finite set if for each k ∈ {1, . . . , m} we have |H ∩ G k | ≥ 1. The blocker of G is the family of all inclusion-wise minimal blocking sets for G.
A family of subsets of a finite set is called a clutter (or a Sperner family) if no set from it contains another. If the family is empty or if it consists of only one subset, {∅}, then the corresponding clutter is called trivial.
The concepts of blocker map and complementary map on clutters [1] made it possible to clarify the relationship between specific families of sets, arising from the matroid theory, and maps on them. The blocker map, that assigns the blocker to a clutter, is defined on all clutters, including trivial clutters.
The following property [2,6] is basic: for a clutter G, the blocker of its blocker coincides with G.
We show that the concepts of blocking set and blocker can be extended when passing from discussing clutters, considered as antichains of the Boolean lattice of all subsets of a finite set, to exploring antichains of arbitrary finite bounded posets (a poset P is called bounded if it has a unique minimal element, denoted0 P , and a unique maximal element, denoted1 P ).
In Section 2, the notion of intersecter plays a role analogous to that played by the notion of blocking set in the Boolean lattice of all subsets of a finite set. In Section 3, we explore the structure of subposets of intersecters in Cartesian products of posets. In Section 4, some properties of the blocker map and complementary map are shortly discussed. In Section 5, the structure of lattices of generalized blockers is reviewed.

Intersecters and complementers
We refer the reader to [7,Chapter 3] for basic information and terminology in the theory of posets.
For a poset Q, Q a denotes its atom set; min Q and max Q denote the sets of all minimal elements and all maximal elements of Q, respectively; I Q (X) and F Q (X) denote the order ideal and order filter of Q generated by a subset X ⊆ Q, respectively. If x, y are elements of Q and x < y (or x ≤ y) then we write x < Q y (or x ≤ Q y). In a similar way, we denote by ∨ Q the operation of join in a join-semilattice Q, and we denote by ∧ Q the operation of meet in a meet-semilattice Q. We use × to denote the operation of Cartesian product of posets.
For a finite family G of finite sets, its conventional blocker is denoted by B(G).
Throughout P stands for a finite bounded poset with |P | > 1. We start with extending of the concept of blocking set.
Definition 2.1. Let A be a subset of P .
• If A = ∅ and A = {0 P } then an element b ∈ P is an intersecter for A in P if for every a ∈ A − {0 P }, we have • If A = {0 P } then A has no intersecters in P .
• If A = ∅ then every element of P is an intersecter for A in P .
• Every non-intersecter for A in P is a complementer for A in P .
Let L denote a finite Boolean lattice. If A is a nonempty subset of the poset L − {0 L } then an element b ∈ L is an intersecter for A in L if and only if I L (b) ∩ L a is a blocking set for the family {I L (a) ∩ L a : a ∈ A}.
We denote by I(P, A) and C(P, A) the sets of all intersecters and all complementers for A in P , respectively. We consider the sets I(P, A) and C(P, A) as subposets of the poset P . For a one-element set {a} we write I(P, a) instead of I(P, {a}) and C(P, a) instead of C(P, {a}).
We have the partition I(P, A)∪ C(P, A) = P . For a nonempty subset A ⊆ P − {0 P }, the subposets of all its intersecters and complementers are nonempty; indeed, we have I(P, A) ∋1 P and C(P, A) ∋0 P . It follows from Definition 2.1 that for such a subset A, we have I(P, A) = I(P, min A) , C(P, A) = C(P, min A) , (2.2) therefore, in most cases, we may restrict ourselves to considering intersecters and complementers for antichains; further, For all antichains (including the empty antichain) Clearly, the subposet I(P, a) of all intersecters for an element a ∈ P is the order filter F P I P (a) ∩P a , hence, in view of (2.3), equality (2.5) in the following lemma holds.
The subposet of all intersecters for A in P is determined by the following equivalent equalities: Proof. To prove (2.6), note that the inclusion follows from the definition of intersecters.
We are left with proving the inclusion Assume that it does not hold, and consider such an intersecter b for A that b ∈ E∈B({I P (a)∩P a :a∈A}) e∈E F P (e). In this case, the inclusion b ∈ e∈E F P (e) holds not for all sets E from the family B({I P (a) ∩ P a : a ∈ A}), hence there exists such an element a ∈ A that |I P (b)∩I P (a)∩ P a | = 0. Therefore b is not an intersecter for A, but this contradicts our choice of b. Hence, (2.6) holds.
Thus, for every antichain A of the poset P , the subposet of all intersecters for A in P is an order filter of P , that is, I(P, A) = F P min I(P, A) . As a consequence, the subposet C(P, A) of all complementers for A in P is the order ideal I P max C(P, A) .
If A is a subset of the poset P then we call the antichain min I(P, A) the blocker of A in P . We call elements of the blocker min I(P, A) minimal intersecters for A in P , and we call elements of the antichain max C(P, A) maximal complementers for A in P .
The images of intersecters under suitable order-preserving maps are also intersecters: Proposition 2.3. Let P 1 and P 2 be disjoint finite bounded posets with |P 1 |, |P 2 | > 1. Let ψ : P 1 → P 2 be an order-preserving map such that Proof. There is nothing to prove for A 1 = ∅ ⊂ P or A 1 = {0 P 1 }. So suppose that A 1 = ∅ ⊂ P and A 1 = {0 P 1 }. Let b 1 be an intersecter for A 1 . According to Definition 2.1, for all a 1 ∈ A 1 , a 1 > P 10 P 1 , we have |I P 1 (b 1 ) ∩ I P 1 (a 1 ) ∩ P 1 a | ≥ 1, and in view of (2.9), for every atom z 1 ∈ I P 1 (b 1 ) ∩ I P 1 (a 1 ) ∩ P 1 a we have the inclusion the left-hand part of which is nonempty. Hence, for all a 2 ∈ ψ(A 1 ) the inclusion b 1 ∈ I(P, A 1 ) implies that This means that ψ(b 1 ) ∈ I P 2 , ψ(A 1 ) and completes the proof.

Intersecters in Cartesian products of posets
In this section, we study the structure of subposets of intersecters in Cartesian products of two finite posets.
Proposition 3.1. Let P 1 and P 2 be disjoint finite bounded posets with where0 Q and1 Q are the adjoint new least and greatest elements. Let A be a nonempty subset of the poset Q − {0 Q ,1 Q }, and let A⇂ P 1 and A⇂ P 2 denote the subsets {a 1 ∈ P 1 : (a 1 ; a 2 ) ∈ A} and {a 2 ∈ P 2 : (a 1 ; a 2 ) ∈ A}, respectively.
Proof. The atom set Q a of the poset Q is P 1 a × P 2 a , therefore, by (2.5), the subposet of intersecters for A in Q is and the statement follows.
Proof. Since the atom set Q a of the poset Q is {0 1 } × P 2 a ∪ P 1 a × {0 2 } , we have, according to equality (2.5), and the statement follows.

Blocker map and complementary map
Let F(P ) denote the distributive lattice of all order filters (partially ordered by inclusion) of P , and let A(P ) denote the lattice of all antichains of P . For antichains A 1 , A 2 ∈ A(P ), we set in other words, we make use of the isomorphism F(P ) → A(P ): F → min F . We call the least element0 A(P ) = ∅ ⊂ P and greatest element 1 A(P ) = {0 P } of the lattice A(P ) the trivial antichains of P . They are counterparts of trivial clutters. Recall (cf. [4]) that for A 1 , A 2 ∈ A(P ),  In particular, for every a ∈ P , a > P0P , we have b({a}) = I P (a) ∩ P a . We also have The following lemma is a reformulation of (2.4): Definition 2.1 implies the following reciprocity property for intersecters: for every antichain A of P , we have (4.6) In the theory of blocking sets the following fact is basic: (see [2,6]). For any clutter G, B B(G) = G.
This statement may be generalized in the following way: Proof. There is nothing to prove for the trivial blockers B =0 B(P ) = ∅ ⊂ P and B =1 B(P ) = {0 P }. So suppose that B is nontrivial.
Choose an arbitrary antichain A ′ ∈ b −1 (B). With regard to reciprocity property for intersecters, every element of A ′ is an intersecter for the antichain B = b(A ′ ). In other words, for each element a ′ ∈ A ′ we have the inclusion a ′ ∈ I(P, B) = b∈B F P b(b) . Taking this inclusion into account, we assign to the antichain A ′ the antichain  Proof. There is nothing to prove for a trivial blocker B, so suppose that B is nontrivial. Choose two antichains A 1 , A 2 ∈ b −1 (B). According to (4.5), we have the following equalities in the lattice A(P ): {e} . (4.8) The Let C(P ) denote the image of A(P ) under the complementary map. The set C(P ) is equipped, by definition, with the partial order induced by the partial order on the lattice of order ideals of P : for C 1 , C 2 ∈ C(P ) we set C 1 ≤ C(P ) C 2 if and only if I P (C 1 ) ⊆ I P (C 2 ).

Lattice of blockers
In this section, we study the structure of the poset of blockers in P .
Lemma 5.1. The poset B(P ) of blockers in P is a meet-subsemilattice of the lattice A(P ).
Proof. We have to prove that for all B 1 , B 2 ∈ B(P ), it holds B 1 ∧ A(P ) B 2 ∈ B(P ). There is nothing to prove when one of the blockers B 1 , B 2 is trivial. Suppose that both B 1 and B 2 are nontrivial. With the help of Theorem 4.3, we write According to (4.5), we have the following equalities in A(P ): Because the restriction map b| B(P ) is bijective, we see that it is an antiautomorphism of B(P ).
We now summarize the information of this section: We call the lattice B(P ) the lattice of blockers in the poset P . It follows immediately from the definition of the complementary map that its restriction c| B(P ) : B(P ) → C(P ), B → c(B), is an isomorphism of B(P ) into the lattice C(P ).