From Clutters to Matroids

This paper deals with the question of completing a monotone increasing family of subsets Γ of a finite set Ω to obtain the dependent sets of a matroid. Specifically, we provide several natural processes for transforming the clutter Λ of the inclusion-minimal subsets of the family Γ into the set of circuits C(M) of a matroid M with ground set Ω. In addition, by combining these processes, we prove that all the minimal matroidal completions of the family can be obtained.


Introduction
A monotone increasing family of subsets Γ of a finite set Ω is a collection of subsets of Ω such that any superset of a set in the family Γ must be in Γ.All the inclusion-minimal elements of Γ determine a clutter Λ, that is, a collection of subsets of Ω none of which is a proper subset of another.Clutters are also known as antichains, Sperner systems or simple hypergraphs.
A wide variety of examples of monotone increasing families exist, among them the collection of the linearly dependent subsets of vectors in a vector space.A matroid M is a combinatorial object that provides an axiomatic abstraction of linear dependence on a finite set Ω.The minimal dependent sets of a matroid M are called circuits.Therefore, the family of circuits of a matroid M is a clutter.
We say that a clutter is matroidal if it corresponds to the family of circuits of a matroid.Matroidal clutters, as well as "almost matroidal" clutters, play a key role in several situations.For instance, in the context of secret-sharing schemes they become a crucial issue for providing general bounds on the optimal information rate of the scheme (see [4,8]).In the framework of algebraic combinatorics and commutative algebra, other interesting examples can be found that deal with monomial ideals and arithmetic properties of the face ring of simplicial complexes (see [1,9]).
Since in general a clutter is far from being matroidal, it is of interest to know how it can be transformed into a matroidal one.This paper deals with the question of finding the matroidal completions of a clutter .
The outline of the paper is as follows.In Section 2 we recall some definitions and basic facts about clutters and matroids.Several ways to obtain matroidal completions of clutters can be found in Section 3; namely, we present the uniform completions (Proposition 3), the I-completions (Proposition 5), and the T -completions (Proposition 8).In addition, by means of the clutter transformations involved in these processes, a necessary condition for a clutter to be a matroid port is obtained (Proposition 9).Finally, Section 4 is devoted to analyzing the minimal matroidal completions.We characterize the clutters with only one minimal element (Theorem 12), and we show how to obtain all the minimal matroidal completions of any clutter (Theorem 13).

Clutters and matroids
In this section we present the definitions and basic facts concerning families of subsets, clutters and matroids that are used in the paper.(The reader is referred to [6,10] for general references on matroid theory).
Let Ω be a finite set.Observe that if Γ is a monotone increasing family of subsets of Ω, then the collection min(Γ) of its inclusion-minimal elements is a clutter; while if Λ is a clutter on Ω, then the family Λ + = {A ⊆ Ω : A 0 ⊆ A for some A 0 ∈ Λ} is a monotone increasing family of subsets.Clearly Γ = (min(Γ)) + and Λ = min Λ + .So a monotone increasing family of subsets Γ is determined uniquely by the clutter min(Γ), while a clutter Λ is determined uniquely by the monotone increasing family Λ + .
Despite the foregoing, we must take into account the following lemma concerning the relationship between the inclusion and the equality of two clutters Λ 1 and Λ 2 , and the inclusion and the equality of their associated monotone increasing families of subsets Λ + 1 and Λ + 2 .
Lemma 1.Let Λ 1 , Λ 2 be two clutters on a finite set Ω. Then: The converse is not true.

Λ
Proof.The proofs of the statements are a straightforward consequence of the definition of Λ + and of the fact that Λ = min Λ + .So it is only necessary to present an example of clutters with Λ 1 Λ 2 and Λ + 1 ⊆ Λ + 2 .For instance, on the finite set Ω = {1, 2, 3}, let us consider the clutters Λ 1 = {{1, 2}, {2, 3}} and Λ 2 = {{1}, {2, 3}}.Then Λ 1 Λ 2 , while The previous lemma leads us to consider a binary relation defined on the set of clutters on Ω. Namely, if Λ 1 and Λ 2 are two clutters on Ω, then we say that Λ 1 Λ 2 if and It is clear that the binary relation is reflexive and transitive.Besides, from statement (1) of the previous lemma, the relation is antisymmetric.Therefore, the binary relation is a partial order on the set of clutters of Ω.We will use this partial order throughout the paper.
There are many interesting families of clutters that can be considered.However, because of their applications, we are interested in clutters that provide matroids.
Matroids are combinatorial objects that can be axiomatized in terms of their independent sets, bases, circuits, rank function, flats, or hyperplanes.Here we present the definition in terms of circuits.A matroid M is an ordered pair M = (Ω, C) consisting of a finite set Ω, called the ground set of the matroid, and a family C of nonempty subsets of Ω which satisfy the following three conditions: The members of the clutter C are the circuits of the matroid M. We shall often write C(M) instead of C. The dependent sets of the matroid are the supersets of the circuits, that is, the dependent sets of M are the members of C(M) + .Therefore, the set of dependent sets of the matroid is a monotone increasing family of subsets whose inclusion-minimal elements are its circuits.A clutter Λ is said to be a matroidal clutter if it is the set of circuits of a matroid, that is, if there exists a matroid M 0 such that C(M 0 ) = Λ.
Since the set of circuits of a matroid is a clutter on the ground set of the matroid, we can consider the partial order induced by on the set of matroids with ground set Ω. Thereby, if M 1 and M 2 are two matroids with ground set Ω, then we say that M 1 M 2 if and only if C(M 1 ) C(M 2 ) where C(M i ) is the clutter of the circuits of M i .So, M 1 M 2 if and only if every circuit of M 1 contains a circuit of M 2 .In matroid theory this is equivalent to saying that the identity map on Ω is a weak map from the matroid M 1 to the matroid M 2 (see [6,Proposition 7.3.11]).

Matroidal completions of a clutter
The set of circuits of a matroid is a clutter, but there are clutters on a finite set Ω that are not the set of circuits of a matroid with ground set Ω. So, a natural question that arises at this point is to determine how to complete a clutter Λ to obtain a matroid; that is to say, to transform the clutter Λ into a matroidal clutter.
In order to look for matroidal completions, it is important to take into account the dependent sets of the matroid instead of the circuits.This is due to the fact that, as the following example shows, there exist clutters Λ such that Λ C(M) for any matroid M.
The above example leads us to the following definition.Let Λ be a clutter on a finite set Ω, and let M be a matroid with ground set Ω. We say that the matroid M is a matroidal completion of the clutter Λ if Λ ⊆ C(M) + .In other words, M is a matroidal completion of Λ if and only if every subset A ∈ Λ is a dependent set in M. From Lemma 1 we get that M is a matroidal completion of Λ if and only if Λ C(M).We will write Λ M instead of Λ C(M).The set of all the matroidal completions of a clutter Λ is denoted by Mat(Λ), that is Mat(Λ) = {M : Λ M}.Observe that if ∅ ∈ Λ then Mat(Λ) = ∅.So, from now on, throughout the paper we assume that ∅ ∈ Λ if Λ is a clutter.As is shown in the next subsection, this assumption guarantees that Mat(Λ) = ∅ for all clutters.
The aim of this section is to provide three methods in order to obtain matroidal completions of Λ; that is, to obtain matroids M in Mat(Λ).By combining these methods, the minimal matroidal completions will be studied in Section 4.

Uniform completion
The following proposition states that the family of uniform matroids provides matroidal completions of clutters.Recall that if Ω is a finite set of size |Ω| = ω and if m ω is a non-negative integer, then the uniform matroid of rank m on Ω is the matroid U m,ω with ground set Ω and set of circuits Proof.The dependent sets of the uniform matroid U m,ω are those subsets We will say that U s−1,ω is the uniform completion of Λ and it is denoted by U(Λ).
The following example shows that, in general, there are matroids in Mat(Λ) that are not uniform matroids.Moreover, from the example, it follows that in general the uniform completion U(Λ) is not a minimal matroidal completion of Λ.

Completion with intersections: I -transformations
In this subsection we prove that it is possible to transform a clutter Λ into a matroidal clutter by adding intersections of suitable subsets of Λ.To present our result we need to introduce some previous definitions.
Let Λ be a clutter on a finite set Ω.For a subset X ⊆ Ω, we denote by I Λ (X) the intersection of the subsets A in Λ contained in X, (this intersection is the one involved in the characterization of the set of circuits in connected matroids, see [6,Theorem 4.3.2]).We say that a clutter Λ is an The proof of the proposition follows from this equivalence.Indeed, if Λ is not a matroidal clutter, then we get that there exist two different subsets M 2,2 , while the I-matroidal completions of Λ 3 are not comparable, that is, M 3,1 M 3,2 and M 3,2 M 3,1 .

Completion with unions: T -transformations
The aim of this subsection is to present some natural ways to obtain matroidal completions of a clutter Λ, that is, to obtain matroids in Mat(Λ).Unlike in the previous subsection, here we proceed in a recursive way by adding, in each step of the process, some slight modifications of the union of two distinct elements of the clutter.Our result is stated in Proposition 8, and by using these matroidal completions a necessary condition for matroid ports is presented in Proposition 9. Let us start by defining the two elementary transformations involved in the recursive process.
Let Λ be a clutter on a finite set Ω. We define the elementary transformations T (1) (Λ) and T (2) (Λ) of Λ as the clutters: that is, in the first elementary transformation T (1) (Λ), we consider the minimal elements of the family obtained by adding to Λ those subsets that arise from the circuit condition.
• T (2) , where A 1 , A 2 ∈ Λ are different} ; that is, in the second elementary transformation T (2) (Λ), we add to Λ the subsets obtained from the union after removing the intersections I Λ (X) defined in the previous subsection.
In this way we obtain the following tree diagram of T -transformations of the clutter Λ, where we write (i 1 , . . ., i r ) instead of T (i 1 ,...,ir) (Λ): (1, 1) (2, 1) We will say that a clutter Λ is a T -transformation of Λ if it is obtained from Λ in this way, that is, if Λ = T (i 1 ,...,ir) (Λ) for some r-tuple (i 1 , . . ., i r ).The next lemma points out the relationship between two T -transformations, that is, between two clutters of the above diagram.The first statement of the lemma deals with the relationship between clutters in each branch of the diagram, whereas the last two statements deal with the comparison of clutters in a same row of the diagram, that is, the 2 r possible clutters T (i 1 ,...,ir) (Λ).
Proof.From the definitions of the two elementary transformations it follows that if Λ 0 is a clutter on Ω then Λ 0 T (1) (Λ 0 ) and Λ 0 T (2) (Λ 0 ).Therefore, we have that the iteration of the elementary transformations provides a monotone increasing sequence of clutters Λ T (i 1 ) (Λ) 1) is concluded by noticing that there are only a finite number of clutters in a finite set.
Next let us prove statement (2).It is necessary to prove the inequality T (i 1 ,...,i r−1 ,1) (Λ) T (i 1 ,...,i r−1 ,2) (Λ) and, in addition, we must also show that in general this inequality is not an equality.Clearly, to do this it is enough to prove that if Λ 0 is a clutter on the finite set Ω then T (1) (Λ 0 ) T (2) (Λ 0 ) and, in addition, we must show that there are clutters Λ 0 of Ω with T (1)  First let us show that T (1) (Λ 0 ) T (2) (Λ 0 ); that is, we must demonstrate that if X ∈ T (1) (Λ 0 ), then there exists X ∈ T (2) (Λ 0 ) such that X ⊆ X.So let X ∈ T (1) then from the definition of T (2) (Λ 0 ), it follows that there exists X ∈ T (2) (Λ 0 ) with X ⊆ X, as we wanted to prove.Therefore, we may assume now that X = (A 1 ∪ A 2 ) \ {x} where A 1 and A 2 are two distinct members of Λ 0 and where x ∈ A 1 ∩ A 2 .At this point we distinguish two cases.First suppose that x ∈ I Λ 0 (A 1 ∪ A 2 ).In such a case we have that ( \ {x} and from the definition of T (2) (Λ 0 ) we get that there exists X ∈ T (2) Then from the definition of I Λ 0 (A 1 ∪ A 2 ) we get that there exists , then there exists X ∈ T (2) (Λ 0 ) with X ⊆ X.Thus, in both cases we conclude that there exists X ∈ T (2) (Λ 0 ) such that X ⊆ X, that is, T (1) (Λ 0 ) T (2) (Λ 0 ), as we wanted to prove.
Finally it is necessary to demonstrate that if Λ 0 is a matroidal clutter then Λ 0 = T (2) (Λ 0 ).Recall that in the proof of Proposition 5 it was stated that a clutter Λ is a matroidal clutter if and only if and so T (2) (Λ 0 ) = Λ 0 .This completes the proof of the proposition.
In some way, the stable value of the above proposition indicates how far Λ is from being a matroid.For instance, from the above proposition it follows that a clutter Λ is the set of circuits of a matroid with ground set Ω if and only if there exists a sequence I such that T (I) * (Λ) = Λ.A matroid is said to be connected if for every pair of distinct elements of the ground set, there is a circuit containing both.A clutter Λ of a finite set Ω is said to be a matroid port if it corresponds to the set of circuits of a connected matroid containing a fixed point, that is, if there exists a connected matroid N with ground set Ω ∪ {ω 0 }, where ω 0 ∈ Ω, such that Λ = {C \ {ω 0 } : ω 0 ∈ C ∈ C(N )}.In such a case it is said that the clutter Λ is the port of the connected matroid N at the point w 0 .
The clutters in the following example show that the necessary condition of the above proposition is not sufficient, and that there is no analogous result if we use the first elementary transformation instead of the second one.
Let us show that there exists no general result concerning the comparison between two different T -matroidal completions of a clutter.The three clutters in the following example illustrate this fact.

Minimal matroidal completions of a clutter
The set Mat(Λ) of all the matroidal completions of a clutter Λ is a non-empty partially ordered set, the poset of matroids of the clutter Λ.Therefore, the minimal elements of the electronic journal of combinatorics 21(1) (2014), #P1.11 this poset will be the minimal matroidal completions of the clutter.In this section we present two results concerning minimal matroidal completions.The first one (Theorem 12) deals with the number of minimal matroidal completions of a clutter, while in the second (Theorem 13) we focus our attention on how the minimal matroidal completions can be obtained.
In general, the poset Mat(Λ) is not a totally ordered set (see Example 11).Therefore, we do not know how many minimal elements this poset has.Our first result states that the non-matroidal clutters have at least two minimal matroidal completions.
Theorem 12. Let Λ be a clutter.Then, the poset Mat(Λ), has a unique minimal element if and only if Λ is a matroidal clutter.
Proof.If Λ is a matroidal clutter, then there exists a matroid M 0 such that C(M 0 ) = Λ, and hence min Mat(Λ) = {M 0 }.Let us show that the converse is true.So let Λ be a clutter and assume that there exists a matroid M such that min Mat(Λ) = {M}.In such a case, it is necessary to demonstrate that Λ is a matroidal clutter.
To The following result concerns non-matroidal clutters; namely, it states that any minimal matroidal completion of the clutter can be obtained by combining the transformations of the previous section.
Theorem 13.Let Λ be a non-matroidal clutter on a finite set Ω and let M be a minimal element of the poset of matroids Mat(Λ), . Then there is a monotone increasing sequence of clutters Λ = Λ 0 Λ 1 • • • Λ r = C(M) such that for i 1, either Λ i is an I-transformation of Λ i−1 or Λ i is a T -transformation of Λ i−1 .
Proof.It suffices to prove that if Λ is a non-matroidal clutter on Ω, and if N is a matroidal completion of Λ , then either there exists an I-transformation Λ 1 of Λ such that Λ 1 N , or there exists a T -transformation Λ 1 of Λ such that Λ 1 N .
So, let Λ be a non-matroidal clutter on Ω and let N be a matroidal completion of Λ .Let us assume that there exists no T -transformation Λ 1 of Λ with Λ 1 N .In such a case, we must demonstrate that there exists an I-transformation Λ 1 of Λ such that Λ 1 N .
do this we consider the blocker b(Λ) of the clutter Λ.The blocker of the clutter Λ is defined as the clutter b(Λ) = min{B ⊆ Ω : B ∩ A = ∅ for all A ∈ Λ}.It is well known that b(b(Λ)) = Λ (see for instance [6, Proposition 2.1.12]).Thus, if X is a subset of Ω such that X ∩ B = ∅ for all B ∈ b(Λ), then X ∈ Λ + .Let us denote b(Λ) = {B 1 , . . ., B s }.For 1 i s let us consider the matroid M B i with ground set Ω and set of circuitsC(M B i ) = {{x} : x ∈ B i }.Since B i ∈ b(Λ), then A ∩ B i = ∅ for all A ∈ Λ.Thus, Λ M B i ,and therefore M M B i because we are assuming min Mat(Λ) = {M}.Let C ∈ C(M) be a circuit of the matroid M. Since M M B i , there exists a circuit C i ∈ C(M B i ) such that C i ⊆ C, and so C ∩ B i = ∅.Therefore, if C ∈ C(M) then C ∩ B i = ∅ for i = 1, . . ., s.Hence, it follows that C ∈ Λ + because b(Λ) = {B 1 , . . ., B s }.Therefore we have that C(M) ⊆ Λ + , and thus C(M) Λ (see Lemma 1).But the matroid M is a matroidal completion of Λ, so Λ C(M).Therefore Λ = C(M), as we wished to prove.
-the T -matroidal completions of Λ 2 are M