Cyclic flats of a polymatroid

Polymatroids can be considered as"fractional matroid"where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.


Introduction
Cyclic flats of a matroid played an important role in matroid theory. They form a ranked lattice, i.e., a lattice with a non-negative number assigned to lattice elements. Bonin and de Mier in [1] gave a characterization of the ranked lattices arising this way. They quote Sims [11] where the rank function of the embedding matroid is specified explicitly by some convolution-like formula.
Generalizing cyclic flats to polymatroids is not completely straightforward, see [5] and [12] where the same definition of polymatroidal cyclic flats arose as we use in this paper. Our main contribution is a complete characterization of the ranked lattice of cyclic flats of a polymatroid together with singleton ranks. Motivated by Sims' construction [11], the convolution of a ranked lattice and a discrete measure (determined by the singleton ranks) is defined. This definition is an extension of the usual convolution of polymatroids, for details see [9]. Similarly to the matroid case, this convolution recovers the embedding polymatroid for the given ranked lattice and measure. We carefully identify the role of different conditions. Knowing which condition ensures what property allows us to use the convolution to create polymatroids with desired properties. It is illustrated by a simple example. A more substantial application is in [3].
Traditionally, convolution is symmetric. In the last section we propose the convolution of two ranked lattices which is symmetric and falls back to the previous notion when the second lattice is a complete modular polymatroid. In a special case this convolution gives an interesting polymatroid extension. It is open under which general and useful conditions will the convolution be a polymatroid.

Notation
All sets in this paper are finite. A polymatroid M = (f, M ) is a non-negative, monotone and submodular function f defined on the subsets of M . Here M is the ground set, and f is the rank function. A polymatroid is integer if the rank function takes integer values only, and it is a matroid if it is integer, and all singletons have value either zero or one. For a throughout treatment of matroids see [10]. Polymatroids were introduced by Edmonds [4], relevant results on polymatroids can be found, e.g., in [2,7,8].
A polymatroid can be considered as "fractional matroid". Relaxing a combinatorial notion to its fractional version allows different techniques to apply. This not different in case of polymatroids. While tools handling matroids are mainly combinatorial, polymatroids have a nice geometrical interpretation allowing geometrical (and continuity) reasoning.
Following the usual practice, ground sets and their subsets are denoted by capital letters, their elements by lower case letters. The union sign is frequently omitted as well as the curly brackets around singletons, thus abA denotes the set {a, b} ∪ A. The set difference has lower priority than union or intersection, . For a function defined on subsets of a set the usual information-theoretical abbreviations are used. Most notably, we write f (I|K) for f (IK) − f (K).
A (discrete) measure µ on M is an additive function on the subsets of M with µ(∅) = 0. As M is finite, the measure is determined by its value on the singletons as µ(A) = {µ(a) : a ∈ A}.
If M = (f, M ) is a polymatroid, then the measure µ M , or just µ f , is the one defined by the singleton ranks. Submodularity implies A collection of subsets is a lattice L if any two elements Z 1 , Z 2 ∈ L have, in L, a least upper bound -their join -, and a greatest lower bound -their meet -, using the standard subset relation as the ordering. The notation for join and meet are Z 1 ∨ Z 2 and Z 1 ∧ Z 2 , respectively. We write Z 1 < Z 2 if Z 1 is strictly below Z 2 , which is the same as Z 1 ⊂ Z 2 . The lattice elements Z 1 and Z 2 are incomparable if they are different and neither Z 1 < Z 2 nor Z 2 < Z 1 holds. As L is finite, it has a smallest element (the meet of all Z ∈ L), denoted by O L , and a largest element (the join of all Z ∈ L), denoted by I L . The pair (λ, L) is a ranked lattice if the rank function λ assigns non-negative real values to lattice elements. λ is pointed if λ(O L ) = 0, and monotone if Z 1 ≤ Z 2 implies λ(Z 1 ) ≤ λ(Z 2 ). The flat C ⊆ M is cyclic if for all i ∈ C either i is a loop, or f (i|C−i) < f (i), see [5,12]. In particular, the minimal flat (containing only loops) is cyclic. When M is a matroid, this definition of cyclic flats is equivalent to the original one, namely that C is the union of circuits (minimal connected sets), see [1,10].

Flats, cyclic flats
Similar to the matroid case, cyclic flats form a lattice. The proof relies on the following structural property of cyclic flats. Lemma 1. Every flat F contains a unique maximal cyclic flat C ⊆ F ; moreover for all C ⊆ A ⊆ F we have Proof. First we show that F contains a maximal cyclic flat C with the given property, then we show that the maximal cyclic flat inside F is unique.
Start with F 1 = F , and suppose we have defined F j for some j ≥ 1. If there is an element It is clear that each F j is a flat, and when F j = F j+1 then it is cyclic. As it contains C, it must equal C. Proof. This is so as the rank of A is the minimum of by submodularity, thus it is enough to show that for some cyclic flat C equality holds. Let F = cl(A) and C ⊆ F be the maximal cyclic flat in F . Then f (F ) = f (A) = f (AC), and by Lemma 1, Proof. Let C 1 and C 2 be cyclic flats. Then C 1 ∩ C 2 is a flat which contains a unique maximal cyclic flat by Lemma 1 above. This is the largest cyclic flat below C 1 and C 2 . The smallest upper bound of C 1 and C 2 is C = cl(C 1 ∪ C 2 ). Indeed, this is a flat, and we claim that it is also cyclic.

Convolution
Let (λ, L) be a ranked lattice and µ be a (discrete) measure both defined on subsets of M . The convolution of the ranked lattice and the measure, denoted by λ * µ, assigns a non-negative value to each subset of M as follows: In the sequel we write r instead of λ * µ. When L contains all subsets of M and λ is the rank function of a polymatroid, then (1) is equivalent to the usual convolution of two polymatroids, see [7,9].

Conditions
Convolution will be used to recover a polymatroid from the lattice of its cyclic flats. Different conditions on the ranked lattice and the measure ensure different properties of the convolution. Rather than listing them repeatedly, we specify them here. In what follows, (λ, L) is a ranked lattice and µ is a measure, both defined on subsets of the same set.
(Z3) for any two lattice elements Z 1 , Z 2 ∈ L, Condition (Z2) is monotonicity, (Z2 * ) is strict monotonicity. (Z3) is submodularity corrected for the the difference between Z 1 ∩ Z 2 and Z 1 ∧ Z 2 (the meet is always a subset of the intersection). Some remarks are due.
Remark 1. (Z3) trivially holds when Z 1 and Z 2 are comparable, thus it is enough to require it to hold for incomparable Z 1 and Z 2 .

Cyclic flats of polymatroids
For each polymatroid M we define a pair of a ranked lattice and a measure as follows: the ranked lattice is collection of cyclic flats endowed with the polymatroid rank, and the measure is µ = µ M generated by the rank of singletons. By Remark 5, the result in [1] characterizing the lattice of cyclic flats of matroids follows immediately from this theorem. The proof proceeds in two stages. The easy part is Claim 5, which shows that the lattice and the measure coming from a polymatroid satisfy the conditions. The converse follows from the fact that the convolution λ * µ recovers a polymatroid which defines λ and µ. This is proved in Claim 6 using a series of claims and lemmas from Section 4 highlighting the role of each condition. The uniqueness follows from Corollary 2: λ and µ together determine the polymatroid rank function. Proof. The minimal cyclic flat O L is the set of loops, i.e., elements with rank zero. This proves (Z1) and (Z5); (Z4) is a simple consequence of submodularity.
If Z 1 < Z 2 then f (Z 1 ) < f (Z 2 ) as Z 1 is a flat and Z 2 is a proper extension of Z 1 . Suppose by contradiction that f (Z 2 ) − f (Z 1 ) = µ(Z 2 −Z 2 ), and let a ∈ Z 2 −Z 1 . Then f (Z 1 ) + µ(Z 2 −Z 1 a) ≥ f (Z 2 −a) by submodularity, and Finally, (Z3) follows from Lemma 1, since Z 1 ∨ Z 2 = cl(Z 1 ∪ Z 2 ), the ranks are equal f (Z 1 ∨Z 2 ) = f (Z 1 ∪Z 2 ). Let F = Z 1 ∩Z 2 ; it is a flat, and C = Z 1 ∧Z 2 is the maximal cyclic flat contained in F . By Lemma 1 Combining these with the submodularity we get the inequality in (Z3). Proof. By Theorem 7 the convolution is a polymatroid. µ = µ M follows from Lemma 9 c). By Claims 12 and 13 elements of the lattice L are precisely the cyclic flats of M. Finally, Claim 11 says that lattice and polymatroid ranks are equal.

Convolution properties
In this section (λ, L) is a fixed ranked lattice and µ is a measure both defined on subsets of M . The convolution function defined on subsets of M will be denoted by r: Theorem 7. If (Z3) holds, then (r, M ) is a polymatroid.
Proof. First observe that for arbitrary subsets A, B, Z A , Z B of M we have

is in both A and B and not in either Z
Finally, if i is in both sets on the right hand side of (3), then i is in both A and B, and not in Z A neither in Z B , thus i is in both sets on the left hand side.
The convolution (r, M ) is a polymatroid if r is non-negative, monotone, and submodular. Non-negativity is clear from the definition (2) as both λ and µ are non-negative. Let A and B be subsets of M ; showing monotonicity. To check submodularity we use Z A ∧ Z B and Z A ∨ Z B to estimate r(A ∩ B) and r(A ∪ B), respectively, as follows: As (the right hand side is a disjoint union), and (3) gives the required inequality.
Proof. We show that r(AZ) = λ(Z) + µ(A), from here the claim follows. Let r(AZ) = λ(Z ′ ) + µ(AZ−Z ′ ) for some Z ′ ∈ L. Then r(aAZ) is minimal, thus Similarly, r(AZ) is minimal, thus as AZ−Z = A by assumption. Combining them we get which holds only if they are equal. Consequently we have equality in (4) as was required. b) Assume (Z2 * ) and (Z3). For every pair A, Z ∈ L such that A is not below Z the above inequality is strict.
Proof. If A ≤ Z then A ∨ Z = Z, thus the two sides are equal. When Z < A then the inequality (strict inequality) follows from condition (Z2) (condition (Z2 * ), respectively). Finally, if A and Z are incomparable, then apply (Z2) (or (Z2 * )) for A ∧ Z and Z, and (Z3) for A and Z to get Their sum is the claimed inequality. When using (Z2 * ), the first inequality is strict, thus the sum is strict as well.

An example
An illustrative example for using convolution is a proof of Helgason's theorem [6] saying that integer polymatroids are factors of matroids. For other examples see [3]. The meet and join are the union and intersection, and O L is the empty set. Define the rank λ as λ : The measure is the expected one: for a ∈ M i let µ(a) = min{1, f (i)}. It is a routine to check that conditions (Z1), (Z2), (Z3) and (Z4) hold. Let N = (r, N ) be the convolution of the ranked lattice and the measure. It is clearly integer, and by Theorem 7 it is a polymatroid. By Lemma 9 r(a) = µ(a), consequently the rank of singletons is either zero or one, which shows thus N is a matroid. Finally, Claim 11 says r(Z) = λ(Z) for all Z ∈ L, therefore M is a factor of N as required.
Helgason's theorem is a special case of a more general statement. In the construction above each singleton i ∈ M with rank f (i) ≥ 2 is replaced by the free matroid of rank f (i). Actually any matroid with this rank can be used which is an immediate consequence of Theorem 14. Let us proceed with some definitions.
The convolution of two ranked lattices (λ 1 , L 1 ) and (λ 2 , L 2 ) is the function on subsets of I L1 ∪ I L2 defined as A → min Z1,Z2 λ ! (Z 1 ) + λ 2 (Z 2 ) : A ⊆ Z 1 ∪ Z 2 as Z 1 runs over elements of L 1 and Z 2 runs over elements of L 2 . If L 2 is the complete subset lattice and λ 2 is a measure, then this formula is equivalent to (1) from Section 2.3. In the very special case of Theorem 14 the convolution of two ranked lattices defines a polymatroid. It is an interesting problem under which general and useful conditions this remains true. Let Let the convolution of the ranked lattices be (r, M P ). We claim that this is the required extension. As both λ 1 and λ 2 are monotone, the minimum is taken when Z 1 and Z 2 is the smallest possible. Consequently for every A ⊆ M P , Using that f (c) = g(P ), conditions in (5) follow easily. Thus one has to check only that (r, M P ) is a polymatroid. Non-negativity and monotonicity is clear, and submodularity can be shown by a case by case checking depending on which terms in (6) provide the smaller value.