© Hindawi Publishing Corp. EXTENDED BLOCKER, DELETION, AND CONTRACTION MAPS ON ANTICHAINS

Families of maps on the lattice of all antichains of a finite 
bounded poset that extend the blocker, deletion, and contraction 
maps on clutters are considered. Influence of the parameters of 
the maps is investigated. Order-theoretic extensions of some 
principal relations for the set-theoretic blocker, deletion, and 
contraction maps on clutters are presented.


Introduction and preliminary.
Let P be a finite bounded poset of cardinality greater than one.We can define some maps on the lattice of all antichains A(P ) of the poset P that naturally extend the set-theoretic blocker, deletion, and contraction maps on clutters; such maps were considered in [4,5].
A set H is called a blocking set for a nonempty family Ᏻ = {G 1 ,...,G m } of nonempty subsets of a finite set if, for each k ∈ {1,...,m}, it holds |H ∩G k | > 0. The family of all inclusionwise minimal blocking sets for Ᏻ is called the blocker of Ᏻ.We denote the blocker of Ᏻ by Ꮾ(Ᏻ).
A family of subsets of a finite ground set S is called a clutter or a Sperner family if no set from that family contains another.The empty clutter ∅ containing no subsets of S and the clutter { 0} whose unique set is the empty subset 0 of S are called the trivial clutters on S. The set-theoretic blocker map reflects a nontrivial clutter to its blocker, and that map reflects a trivial clutter to the other trivial clutter: Ꮾ(∅) = { 0} and Ꮾ({ 0}) = ∅.
Let X ⊆ S and |X| > 0. The set-theoretic deletion (\X) and contraction (/X) maps are defined in the following way: if Ᏻ is a nontrivial clutter on S, then the deletion Ᏻ\X is the family {G ∈ Ᏻ : |G ∩ X| = 0} and the contraction Ᏻ/X is the family of all inclusionwise minimal sets from the family {G − X : G ∈ Ᏻ}.The deletion and contraction for the trivial clutters coincide with the clutters ∅\X = ∅/X = ∅ and { 0}\X = { 0}/X = { 0}.The maps (\ 0) and (/ 0) are the identity map on clutters; for any clutter Ᏻ, we by definition have Ᏻ\ 0 = Ᏻ/ 0 = Ᏻ.
Let Ᏻ be a clutter on the ground set S. Given a subset X ⊆ S, we have Recall that the atoms of the poset P are the elements covering its least element.Let X be a subset of the atom set P a of P .(We denote the empty subset of P a by ∅ a .)We use the denotation b : A(P ) → A(P ) for the order-theoretic blocker map from [4], and we use the denotations (\X), (/X) : A(P ) → A(P ) for the order-theoretic operators of deletion and contraction from [5], respectively.We do not recall those concepts here because the map b is the (∅ a , 0)-blocker map from Definition 2.1 of the present paper and the maps (\X) and (/X) are the (X, 0)-deletion and (X, 0)-contraction maps from Definition 3.1 of the present paper, respectively.
In the present paper, we consider families of the so-called (X, k)-blocker, (X, k)-deletion, and (X, k)-contraction maps on A(P ) parametrized by subsets X ⊆ P a and numbers k ∈ N, k < |P a |.We show that for all pairs of the abovementioned parameters X and k, the essential properties of the maps remain similar to those of the (∅ a , 0)-blocker, (X, 0)-deletion, and (X, 0)-contraction maps on A(P ) that were investigated in [4,5].In particular, we present analogues of relations (1.3) and (1.4) in Proposition 2.6(ii) and Theorem 3.7.
We refer the reader to [7,Chapter 3] for basic information and terminology in the theory of posets.
We use min Q to denote the set of all minimal elements of a poset Q.If Q has a least element, then it is denoted 0Q ; if Q has a greatest element, then it is denoted 1Q .
Throughout the paper, P stands for a finite bounded poset of cardinality greater than one, that is, P by definition has the least and greatest elements that are distinct.We denote by I(A) and F(A) the order ideal and filter of P generated by an antichain A, respectively.
All antichains of P compose a distributive lattice denoted A(P ); in the present paper, antichains are by definition partially ordered in the following way; if A ,A ∈ A(P ), then we set We call the least and greatest elements 0A(P) and 1A(P) of A(P ) the trivial antichains of P because, in the context of the present paper, they are counterparts of the trivial clutters.Here, 0A(P) is the empty antichain of P and 1A(P) the oneelement antichain { 0P }.We denote by ∨ and ∧ the operations of join and meet in the lattice A(P ); if A ,A ∈ A(P ), then 2. (X, k)-blocker map.In this section, we consider a family of maps on antichains of a finite bounded poset that extend the set-theoretic blocker map on clutters.From now on, X is always a subset of P a and k is a nonnegative integer less than |P a |.
if A is nontrivial, and We use the denotations b k and b X instead of the denotations b ∅ a k and b X 0 , respectively.The (∅ a , 0)-blocker map is the blocker map b on A(P ) considered in [4].Given A ∈ A(P ), the antichain b(A) is called the blocker of A in P .
If {a} is a one-element antichain of P , then we write b X k (a) instead of b X k ({a}).Let a ≠ 0P .Since the blocker map on A(P ) is antitone, for every The following statement immediately follows from Definition 2.1.

Lemma 2.2. Let A be a nontrivial antichain of
Let a ∈ P , a ≠ 0P .From now on,a denotes the family of subsets of the atom set P a defined as follows: (2.4) Let L(P a ) denote the Boolean lattice of all subsets of the atom set P a , and let L(P a ) (k+1) denote the subset of all elements of rank k + 1 of L(P a ).Given a (k + 1)-subset E ⊆ P a , we denote by ε(E) the least upper bound for E in L(P a ); conversely, given an element e ∈ L(P a ) (k+1) , we denote by ε −1 (e) the (k + 1)subset of all atoms of L(P a ) that are comparable with e.
Let A be a nontrivial antichain of P .If |b(a) − X| ≤ k for some a ∈ A, then Definition 2.1 implies b X k (A) = 0A(P) .In the case |b(a) − X| > k for all a ∈ A, Proposition 2.3 describes two alternative ways of elementwise finding the (X, k)-blocker of A; it involves the set-theoretic blocker Ꮾ(•) of a set family. (2.5) Proof.We have and an order-theoretic argument shows that, for every a ∈ A, it holds that assume that it does not hold.Consider an element b ∈ b X k (A) such that it does not belong to the right-hand side of (2.8).In this case, there is an element a ∈ A such that |I(b) ∩ I(a) ∩ (P a − X)| ≤ k.It means that the left-hand side of (2.8) is not an (X, k)-blocker of A, a contradiction.
The following lemma clarifies how the parameters of the (X, k)-blocker map influence the image of A(P ); additionally, the lemma states that b X k is antitone.
Lemma 2.4.(i) Let Y ⊆ P a , Y ⊇ X, and let j be a nonnegative integer, j ≤ k. (2.10)

Proof. (i)
There is nothing to prove if A is trivial.Suppose that A is a nontrivial antichain of P .For each element a ∈ A, we by (2.7) have With respect to (2.6), this yields (2.12) The relation b X j (A) ≥ b X k (A) is proved in a similar way.(ii) If A is a trivial antichain, then the assertion immediately follows from Definition 2.1.Suppose that A is nontrivial.For every a ∈ A , there is a ∈ A such that {a } ≤ {a } and, as a consequence, it holds the inclusion b(a , and the proof is completed by applying (2.6).
In addition to Lemma 2.4(ii), we need the following statement to describe the structure of the image of A(P ) under the (X, k)-blocker map.

.13)
Proof.If A is a trivial antichain of P , then the lemma follows from Definition 2.1 because, in this case, we have b ≥ A and we are done.Finally, suppose that b X k (A) is a nontrivial antichain.On the one hand, according to Lemma 2.2, for each a ∈ A and for all b ∈ b X k (A), it holds that On the other hand, we, by Definition 2.1, have We complete this section by applying a standard technique of the theory of posets to the lattice A(P ) and the (X, k)-blocker map on it.See, for instance, [1, Chapter IV] on (co)closure operators.
The lattice B X k (P ) is a meetsubsemilattice of the lattice A(P ).
(iii) For every B ∈ B X k (P ), its preimage (b X k ) −1 (B) under the (X, k)-blocker map is a convex join-subsemilattice of the lattice A(P ).The greatest element of (b Proof.In view of Lemmas 2.4(ii) and 2.5, assertions (i) and (ii) are a corollary of [1,Propositions 4.36

and 4.26]. To prove (iii), choose arbitrary elements
We call the poset B X k (P ) from Proposition 2.6(ii) the lattice of (X, k)-blockers in P .The poset B(P ) = B ∅ a 0 (P ) is called in [4] the lattice of blockers in P .

(X, k)-deletion and (X, k)-contraction maps.
In this section, we consider order-theoretic extensions of the set-theoretic deletion and contraction maps on clutters.Definition 3.1.(i) If {a} is a nontrivial one-element antichain of P , then the (X, k)-deletion {a}\ k X and (X, k)-contraction {a}/ k X of {a} in P are the antichains (iii) The (X, k)-deletion and (X, k)-contraction of the trivial antichains of P are 0A(P) \ k X = 0A(P) / k X = 0A(P) , (iv) The map is the operator of (X, k)-deletion on A(P ).
The map is the operator of (X, k)-contraction on A(P ).
Given an antichain A ∈ A(P ), we use the denotations A\X and A/X instead of the denotations A\ 0 X and A/ 0 X, respectively.The (X, 0)-deletion map (\X) : A(P ) → A(P ) and the (X, 0)-contraction map (/X) : A(P ) → A(P ) are the operators of deletion and contraction on A(P ), respectively, considered in [5].
The following observation is an immediate consequence of Definition 3.1.If a ,a ∈ P and {a } ≤ {a } in A(P ), then hence, in view of (3.3) and (3.4), we can formulate the following lemma.

Lemma 3.2. If A ,A ∈ A(P ) and A ≤ A , then
Moreover, if {a} is a one-element antichain of P , then we have and a more general statement is true.
Another consequence of Definition 3.1 is that, for a one-element antichain Let {a} be a nontrivial one-element antichain of P .We obviously have ({a}\ k X)\ k X = {a}\ k X.We show that ({a}/ k X)/ k X = {a}/ k X.If |b(a) ∩ X| ≤ k, then Definition 3.1 implies ({a}/ k X)/ k X = {a}/ k X = {a}; further, if |b(a)∩X| > k and b(a) ⊆ X, then Definition 3.1 implies ({a}/ k X)/ k X = {a}/ k X = 1A(P) .Suppose that |b(a) ∩ X| > k and b(a) ⊆ X.In this case, on the one hand, we have ({a}/ k X)/ k X ≥ {a}/ k X by Lemma 3.3, on the other hand, for every element , and, as a consequence, we have We arrive at the conclusion that ({a}/ k X)/ k X = {a}/ k X.With respect to (3.3), we can formulate the following lemma.
(3.12) Lemmas 3.2, 3.3, and 3.4 lead to a characterization of the (X, k)-deletion and (X, k)-contraction maps in terms of (co)closure operators.
Proposition 3.5.The map (\ k X) is a coclosure operator on A(P ).The map (/ k X) is a closure operator on A(P ).
The following proposition is a counterpart of Lemma 2.4(i).
Proof.If A is a trivial antichain, then the proposition follows from (3.4).Suppose that A is nontrivial.For each a ∈ A, (3.1) Other relations are proved in a similar way.
We denote the images (\ k X)(A(P )) = {A\ k X : A ∈ A(P )} and (/ k X)(A(P )) = {A/ k X : A ∈ A(P )} by A(P )\ k X and A(P )/ k X, respectively.We can interpret well-known properties of (semi)lattice maps and (co)closure operators on lattices in the case of the (X, k)-deletion and (X, k)-contraction maps.
To prove b X k (A)/ k X ≤ b X k (A\ k X), we use (3.18) and (3.11), and we see that (3.20)

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Proposition 3 . 6 .
Let Y ⊆ P a , Y ⊇ X, and let m be an integer, k ≤ m < |P a |.If A ∈ A(P ), then