Lattices related to extensions of presentations of transversal matroids

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; we order these extensions by the weak order. We show that $T_{\mathcal{A}}$ is a distributive lattice, and that each finite distributive lattice is isomorphic to $T_{\mathcal{A}}$ for some presentation $\mathcal{A}$ of some transversal matroid $M$. We show that $T_{\mathcal{A}}\cap T_{\mathcal{B}}$, for any two presentations $\mathcal{A}$ and $\mathcal{B}$ of $M$, is a sublattice of both $T_{\mathcal{A}}$ and $T_{\mathcal{B}}$. We prove sharp upper bounds on $|T_{\mathcal{A}}|$ for presentations $\mathcal{A}$ of rank less than $r(M)$ in the order on presentations; we also give a sharp upper bound on $|T_{\mathcal{A}}\cap T_{\mathcal{B}}|$. The main tool we introduce to study $T_{\mathcal{A}}$ is the lattice $L_{\mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of $\{1,2,\ldots,r(M)\}$.


INTRODUCTION
We continue the investigation, begun in [4], of the extent to which a presentation A of a transversal matroid M limits the single-element transversal extensions of M that can be obtained by extending A. The following analogy may help orient readers. A matrix A, over a field F, that represents a matroid M may contain extraneous information; this can limit which F-representable single-element extensions of M can be represented by extending (i.e., adjoining another column to) A. For instance, for the rank-3 uniform matroid U 3, 6 , partition E(U 3,6 ) into three 2-point lines, L 1 , L 2 , and L 3 . Let A be a 3 × 6 matrix, over F, that represents U 3,6 . The line L i is represented by a pair of columns of A, which span a 2-dimensional subspace V i of F 3 . While V i ∩ V j , for {i, j} ⊂ {1, 2, 3}, has dimension 1 (since the corresponding lines of U 3,6 are coplanar), the intersection V 1 ∩ V 2 ∩ V 3 can, in general, have dimension either 0 or 1: this dimension is extraneous. If dim(V 1 ∩ V 2 ∩ V 3 ) is 1, then no extension of A represents the extension of M that has an element on, say, L 1 and L 2 but not L 3 ; otherwise, no extension of A represents the extension of M that has a non-loop on all three lines. (The underlying problem is the lack of unique representability, which is a major complicating factor for research on representable matroids. See Oxley [12,Section 14.6].) In this paper, we consider such problems, but for transversal matroids in place of F-representable matroids, and presentations in place of matrix representations.
A transversal matroid can be given by a presentation, which is a sequence of sets whose partial transversals are the independent sets. In [4], we introduced the ordered set T A of transversal single-element extensions of M that have presentations that extend A (i.e., the new element is adjoined to some of the sets in A), where we order extensions by the weak order. In Section 3, we introduce a new tool for studying T A : given a presentation A of a transversal matroid M with the number, |A|, of terms in the sequence A being the rank, r, of M , we define a closure operator on the lattice 2 [r] of subsets of the set [r] = {1, 2, . . . , r}, and we show that the resulting lattice L A of closed sets is a (necessarily Extending a presentation A = (A i : i ∈ [r]) of a transversal matroid M consists of adjoining an element x that is not in E(M ) to some of the sets in A. More precisely, for an element x ∈ E(M ) and a subset I of [r], we let A I be (A I i : i ∈ [r]) where The matroid M [A I ] on the set E(M ) ∪ {x} is a rank-preserving single-element extension of M . (This is the only type of extension we consider, so below we omit the adjectives "rank-preserving" and "single-element".) Throughout this paper, we reserve x for the element by which we extend a matroid. We will use principal extensions of matroids, which we now recall. Mason [11] showed that if (A i : i ∈ [r]) and (B i : i ∈ [r]) are maximal presentations of the same transversal matroid, then there is a permutation τ of [r] with A τ (i) = B i for all i ∈ [r]. (Minimal presentations, in contrast, are often more varied.) The next lemma, which is due to Bondy and Welsh [2] and plays important roles in this paper, gives a constructive way to find the maximal presentations of a transversal matroid. (1) the set system obtained from A by replacing A i by A i ∪ {e} is also a presentation of M , and (2) e is a coloop of the deletion M \A i .
A routine argument shows that the complement E(M ) − A i of any set A i in A is a flat of M [A]. By Lemma 2.6, the complement of each set in a maximal presentation of M is a cyclic flat of M . Bondy and Welsh [2] and Las Vergnas [10] proved the next result about the sets in minimal presentations.
. The next result, by Brualdi and Dinolt [6], follows from the last two lemmas.
Corollary 2.9. The ordered set of presentations of a rank-r transversal matroid M is ranked; the rank of a presentation This corollary applies to both the order we focus on, A B, and the more customary order, A ≤ B; the rank of a presentation is the same in both orders.
The weak order ≤ w on matroids on the same set E is defined as follows: M ≤ w N if r M (X) ≤ r N (X) for all subsets X of E; equivalently, every independent set of M is independent in N . This captures the idea that N is freer than M . The next two lemmas are simple but useful observations. Lastly, we recall how to think of transversal matroids geometrically and to give affine representations of those of low rank, as in Figures 1 and 2 ) on E can be encoded by a 0-1 matrix with r rows whose columns are indexed by the elements of E in which the i, e entry is 1 if and only if e ∈ A i . If we replace the 1s in this matrix by distinct variables, say over R, then it follows from the permutation expansion of determinants that the linearly independent columns are precisely the partial transversals of A, so this is a matrix representation of M [A]. One can in turn replace the variables by nonnegative real numbers and preserve which square submatrices have nonzero determinants; one can also scale the columns so that the sum of the entries in each nonzero column is 1. In this way, each non-loop of M is represented by a point in the convex hull of the standard basis vectors. This yields the following geometric picture: label the vertices of a simplex 1, 2, . . . , r and think of associating A i to the i-th vertex, then place each point e of E freely (relative to the other points) in the face of the simplex spanned by s A (e).

A CLOSURE OPERATOR AND TWO ISOMORPHIC DISTRIBUTIVE LATTICES
Let A be a presentation of M . In [4], we introduced the ordered set T A of transversal extensions of M that have presentations that extend A, ordering T A by the weak order. As the results in this paper demonstrate, the lattice L A of subsets of [r(M )] that we define in this section and show to be isomorphic to T A is very useful for studying T A .
Recall that we consider only single-element rank-preserving extensions. Also, x always denotes the element by which we extend a matroid.
3.1. The lattice L A . The first lattice we discuss is the lattice of closed sets for a closure operator that we introduce below, so we first recall closure operators (see, e.g., [1, p. 49]). A closure operator on a set S is a map σ : 2 S → 2 S for which (1) X ⊆ σ(X) for all X ⊆ S, (2) if X ⊆ Y ⊆ S, then σ(X) ⊆ σ(Y ), and (3) σ(σ(X)) = σ(X) for all X ⊆ S.
Given a closure operator σ : 2 S → 2 S , a σ-closed set is a subset X of S with σ(X) = X. The set of σ-closed sets, ordered by containment, is a lattice; join and meet are given by X ∨ Y = σ(X ∪ Y ) and X ∧ Y = X ∩ Y . By property (1), the set S is σ-closed. define a map σ A : 2 [r] → 2 [r] by setting σ A (I) = K. We next show that σ A is a closure operator. We use L A to denote the lattice of σ A -closed sets. See Figure 1 for examples. Let I and J be in L A . Their meet, I ∧ J, is I ∩ J since, as noted above, this holds for any closure operator. We claim that I ∨ J = I ∪ J.
We now show how the order on presentations relates to the lattices of closed sets.
The corollary below is a theorem from [4].    It is easy to check that both T A and T B consist of just the extension by a loop, U 3,4 ⊕ U 0,0 , and the free extension, U 3,5 . Thus, From the next result, which is a reformulation of [4, Theorem 3.1], we see that we cannot recover the presentation A from L A .
As Example 1 shows, we cannot always reconstruct the sets in A from T A ; however, in some cases we can. For the matroid in Figure 1, one can check that the sets in each of its presentations A can be reconstructed from T A . Also, as we now show, for any transversal matroid M , the sets in each minimal presentation A of M can be reconstructed from T A . By Theorem 3.6, from Proof. Set r = r(M ). First assume that X satisfies condition (1).
show that all sets in L A are given by items (1) and (2), it suffices to show that for each h Item (1 ′ ) can replace item (1) since, by Lemma 2.4, r(Y ) = |s A (Y )| for a set Y if and only if |X| = |s A (X)| for some (equivalently, every) basis X of M |Y . By Lemma 2.5, in terms of T A , the extension that corresponds to a set s A (X) in item (1) of Theorem 3.7 is the principal extension, M + X e.
Proof. The first assertion follows from Theorem 3.7 since cyclic flats satisfy condition (1 ′ ). Now let A be maximal. By Theorem 3.7, it suffices to show that if X is an independent set The next result identifies some closed sets in terms of known closed sets and supports.
For the last assertion, take J = s A (e) − {h} and F = {e}.
The next result gives conditions under which the support of a set is, or is not, closed.
Proof. We start with an observation.  Figure 2 shows. Each presentation A of M is both maximal and minimal, so L A = 2 [4] . However, {2, 3} is not an intersection of the A-supports of singletons. Thus, the sets s A (e) generate L A , but both their unions and the intersections of such unions are needed to obtain all of L A .     is M [B J1∪J2 ]. As claimed, these matroids are equal since, by Lemma 3.12, where ∨ denotes the join in the lattice of extensions of M .
The situation for meets is more complex, as the example below illustrates.  Figure 3, is not transversal. One way to see this is that the three coplanar 3point lines through x are incompatible with the affine representation described at the end of Section 2. That view also implies that the meet of M 1 and M 2 in both T A and T B is formed by extending M by a loop.
This example illustrates the next result: the meet of M 1 and M 2 in T A is their meet in T B (even though these can differ from their meet in the lattice of all extensions). The proof of this theorem uses the following result from [4].
Proof of Theorem 3.14. The closure of L A,B under unions follows from the argument that gives equation (3.2). We next show that the closure of L A,B under intersections follows from statement (3.14.1), which we then prove. (3.14.1) For subsets X 1 , X 2 , . . . , and where C x is the set of circuits of M ′ that contain x. Now We claim that for each k ∈ [t], we have To see this, let cl be the closure operator of M , and cl I that of M [A I ]. For any y ∈ C−{x}, Thus, y ∈ cl (C − {x, y}) ∪ X k . By the formulation of closure in terms of circuits (as in [12, Proposition 1.4.11]), it follows that each y ∈ C − (X k ∪ {x}) is in some circuit, say C y , of M with C y ⊆ X k ∪ (C − {x}). Now |s A (C y )| = r(C y ) = |s B (C y )| by Lemma 2.4. Since this applies for each y ∈ C − (X k ∪ {x}), and since we also have |s A (X k )| = r(X k ) = |s B (X k )|, equation ( for any non-empty subset P of [t]. Thus, for any such P , The same argument applies to B and gives The deductions in the previous two paragraphs and inclusion-exclusion give The assertions about L B,A and T A ∩ T B now follow easily. The proof of Theorem 3.14 and its reduction to statement (3.14.1) give the following alternative description of L A,B .
The sets I that satisfy condition ( * ) correspond to the principal extensions M + X x of M that are common to T A and T B .
We conclude this section with two corollaries. Note that we can iterate the operation of extending set systems to get (A I1 ) I2 , where x 1 is added in A I1 , and x 2 is added in (A I1 ) I2 . We next show that such extensions, using sets in L A,B , are compatible. Proof. The result follows from two observations: (i) Theorem 3.7 yields I 2 ∈ L A I 1 and J 2 ∈ L B J 1 ; (ii) if I 2 and X satisfy condition ( * ) above in M , then so do I 2 and X in M [A I1 ], and likewise for intersections of sets that satisfy condition ( * ). Proof. Apply Corollary 3.17 repeatedly, with each I h = I and each J h = J, until the set of added elements is cyclic in the extension; the rank of this cyclic set must be both |I| and |J|.
3.5. How to get any finite distributive lattice. We show that each sublattice of 2 [r] that includes both ∅ and [r] is the lattice L A for some presentation A of some transversal matroid of rank r; indeed, we prove two refinements of this result. Up to isomorphism, this result covers all finite distributive lattices since each such lattice L is isomorphic to the lattice of order ideals of some finite ordered set (specifically, the induced order on the set of join-irreducible elements of L; see, e.g., [1,Theorem II.2.5]). Combining the result below with Theorem 3.4 shows any distributive lattice is isomorphic to T A for some presentation A of some transversal matroid. Proof. To prove assertion (1), for each non-empty set I ∈ L, let X I be a set of |I| + 1 elements that is disjoint from all other such sets X J . For i with 1 ≤ i ≤ r, let Thus, if e ∈ X I , then s A (e) = I. The presentation A of M is maximal since, with |X I | > |I| and s A (X I ) = I, the set X I is dependent in M , yet if we adjoin any element of X I to any set A j with j ∈ I, then the resulting set system A ′ has a matching of X I , so X I is independent in M [A ′ ]. It now follows from Theorem 3.10 that L ⊆ L A . Since L and L A are sublattices of 2 [r] and s A (e) ∈ L for all e ∈ E(M ) by construction, we get s A (F ) ∈ L for each cyclic flat F of M , so Corollary 3.8 gives L A ⊆ L. Thus, L A = L. Figure 4 illustrates the proof of assertion (2). Let [n] be the ground set of U r,n . For I ∈ L, let I 0 be the (possibly empty) set of elements that occur first in I, that is, Since L is closed under intersection, for each i ∈ [r], there is exactly one I ∈ L with i ∈ I 0 ; using that I, set The irreducible elements of a finite distributive lattice L are of great interest. The order induced on the set of join-irreducibles of L is isomorphic to that induced on its set of meet-irreducibles, and the lattice of order ideals of each of these induced suborders of L is isomorphic to L itself. (See, e.g., [1, Theorem II.2.5 and Corollary II.2.7].) Thus, the rank of L is the number of join-irreducibles in L, which is also its number of meet-irreducibles.
We now study the irreducible elements of the lattices L A introduced above. The least set S i in L A that contains a given element i ∈ [r] is J∈LA : i∈J J. The sets S i are not limited to the atoms of L A ; see the examples in Figure 1. Clearly S i is joinirreducible. Each set U in L A is i∈U S i , so there are no other join-irreducibles of L A . Thus, the number of join-irreducibles is the number of distinct sets S i . Note that if A i and A j in A are equal, then S i = S j since, for X ⊆ E(M ), we have i ∈ s A (X) if and only if j ∈ s A (X). Thus, the number of join-irreducible sets in L A is at most the number of distinct sets in A. As Example 1 shows, this bound can be strict (there, A has three distinct sets but L A has only one join-irreducible; likewise for B).
The greatest set in L A that does not contain a given element i ∈ [r] is J∈LA : i ∈J J. An argument like that above, or an application of order-duality, shows that these are the meetirreducibles of L A . By the remark after the proof of Theorem 3.7, each meet-irreducible element of L A corresponds to a principal extension of M ; the converse is false, since for instance, in either example in Figure 1, the set {2, 3} corresponds to a principal extension, but {2, 3} is the meet of the sets {1, 2, 3} and {2, 3, 4} in L A .
We now identify a join-sublattice L ′ A of L A that, by Theorem 3.7, has the same the meet-irreducibles, thereby reducing the problem of finding the meet-irreducibles of L A to the same problem on a potentially smaller lattice. Set (Adding the condition that X is independent would not change L ′ A .) By Theorem 3.7, L ′ A ⊆ L A and L ′ A generates L A since L A consists precisely of the intersections of the sets in L ′ A . Lemma 3.15 shows that L ′ A is a join-sublattice of L A . Each lattice is isomorphic to L ′ A for a maximal presentation A of some transversal matroid (see the proof of [3, Theorem 2.1]). By Corollary 3.8, when the presentation A is maximal, the same conclusions hold for the (often smaller) lattice

APPLICATIONS
Theorems 4.1 and 4.5 below are applications of the results in Section 3. Both results stem from the observation that proper sublattices of 2 [r] must be substantially smaller than 2 [r] . (The special case of maximal proper sublattices of 2 [r] have been studied in other settings, such as finite topologies; see, e.g., Sharp [14] and Stephen [15].) We first give examples to show that, for 1 ≤ i < r, the bounds are sharp.
Thus, s A k (e) = [k + 1]. Each A k is a presentation of M , the presentation A 0 is minimal, and A k−1 ≺· A k for k ≥ 1. Thus, A k has rank k in the ordered set of presentations. where X denotes the complement of the set X. From the first description of L A k , we get The proof of the bound in Theorem 4.1 uses Lemma 4.3, which catalogs the sublattices of 2 [r] that have more than 2 r−1 elements. The proof of that lemma uses the following result by Chen, Koh, and Tan [7] (see the proof in Rival [13]).
16 · 2 r . Also, L V is not contained in any sublattice L of 2 [r] with |L| = 5 8 · 2 r . Proof. To prove this result, we apply Lemma 4.2 recursively. To simplify the argument, note that U ′ i is the image of U i under the complementation map X → X (which is orderreversing) of 2 [r] ; this allows us to pursue only the lattices L V and L 1 , L 2 , . . . , L r−1 below.
The join-irreducibles of 2 [r] are the singleton sets, and the meet-irreducibles are their complements, so by Lemma 4.2, the maximal proper sublattices of 2 [r] are L 1 and its images under permutations of [r] (the lattice L ′ 1 is obtained by such a permutation). To verify the assertions below about join-irreducibles, note that (i) each join-irreducible of L i−1 that is also in L i is join-irreducible in L i , and (ii) L i has at most r join-irreducibles. To complete the proof, we induct to show that for i with 3 ≤ i < r, the only maximal proper sublattice L of L i−1 with |L| > 2 r−1 is L i , up to permuting elements. We include the following conditions in the induction argument (see Figure 5): (i) the join-irreducibles of L It is easy to check that conditions (i) and (ii) hold for L i , which completes the induction.
The last background item we need before proving the upper bounds in Theorem 4.1 is the following lemma from [4]. Proof of Theorem 4.1. Consider presentations A 0 ≺· A 1 ≺· · · · ≺· A r of M where A 0 is minimal. Thus, A j has rank j in the order on presentations, and L A j is a sublattice of L A j−1 . By Lemma 4.3, if |L A j | > 2 r−1 , then |L A j | = 1 2 + 1 2 i+1 2 r for some i with 1 ≤ i < r, so it suffices to prove the following statement: if |L A j | = 1 2 + 1 2 i+1 2 r , then j ≤ i. For i = 1, assume |L A j | = 3 4 · 2 r . By Lemma 4.3, up to permuting [r], we have (2) of Corollary 3.11 holds (h is 1), so L A j is properly contained in L A j−1 ; since L A j is a proper sublattice only of 2 [r] , we have L A j−1 = 2 [r] . Thus, A j−1 is a minimal presentation by Theorem 3.6, so j − 1 = 0, so j = 1.
For Condition (2) of Corollary 3.11 holds (h is 1 in the first case and either 2 or 3 in the second), so L A j is properly contained in L A j−1 . Thus, |L A j−1 | ≥ 3 4 · 2 r . The previous case gives j − 1 ≤ 1, so j ≤ 2.
The general case with L A j = L i or L A j = L ′ i follows inductively in the same manner. We turn to the only case that requires a more involved argument, namely Since A j−1 ≺· A j , we have s A j−1 (e) s A j (e) for some e ∈ E(M ), so s A j−1 (e) ∈ L V by Theorem 3.  (1) of Corollary 3.11; thus, |L A j−1 | ≥ 3 4 · 2 r , so j − 1 ≤ 1, so j < 3. We may now assume that L A j = L A j−1 and that s First assume that for all options for the terms A 0 , A 1 , . . . , A j−1 , the only element d with s A j (d) = s A k (d) for some k < j is d = e. Lemma 4.4 then implies that e is a coloop of M ; also, the presentation of M \e that is obtained by removing e from all sets in A 0 is minimal. This case is covered by the example that we used to show that the bound is sharp, so we may now assume that e is not a coloop of M .
Let A and B be presentations of M . In Theorem 3.14 we showed that T A ∩ T B is a sublattice of both T A and T B . The smallest that |T A ∩ T B | can be is two, with these two common extensions being the free extension and the extension by a loop; for instance, the two minimal presentations  4 · 2 r . This bound is sharp. Proof. The inequality follows from Theorems 4.1 and 3.14 if either A or B is not minimal, so we may assume that both are minimal. As shown in Section 3.2, when A is minimal, we can reconstruct the sets in A from T A ; thus, by our assumption, T A = T B , so L A,B is a proper sublattice of L A . Thus, we get the bound by our work above.

ACKNOWLEDGMENTS
The author thanks Anna de Mier for very useful feedback on the ideas in this paper, for comments that improved the exposition, for catching a flaw in the original proof of Theorem 3.14, and for observations that led to Theorem 3.10.