Separations of sets
Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied.
We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.
Separations of graphs have been studied ever since Robertson and Seymour introduced the notion of tangles of graphs in . Tangles allow one to describe highly cohesive regions and objects in a graph not directly, say by listing the vertices belonging to that region or object, but rather indirectly by simply stating for each of the graph’s low order separations which of the two sides of that separation (most of) the region lies on. The upside of this approach is that the highly cohesive region can be described in this way even if it is a little fuzzy – e.g., when every individual vertex or edge lies on the wrong side of some low-order separation, such as in the case of a large grid: for every low-order separation, almost all of the grid will lie on one side of it, giving rise to a ‘consistent’ orientation of the low-order separations, but every individual vertex of the grid lies on the ‘other’ side of the separator consisting of just its four neighbours.
In , abstract separation systems were introduced to axiomatize, and thereby generalize, this notion of separations and tangles in graphs, so as to make it applicable to cohesive structures also in matroids and other combinatorial objects. This general framework is flexible enough to deal with a plethora of different applications (see [3, 4, 5] for more). Furthermore, some central structure theorems about tangles can be proved in this setting and then applied to more specific applications, for instance to obtain an elementary proof of Robertson and Seymour’s duality theorem connecting tree-width and brambles, which started the study of tangles in .
While treating separations at this abstract level can make the behaviour of their concrete instances more transparent, we need those concrete types of separation to guide our intuition also when we study abstract separation systems. We are therefore led to consider the representation problem familiar from other algebraic contexts: Which abstract separation systems can be represented as separations of graphs? Or as separations of sets such as set bipartitions?
This paper seeks to answer these questions by giving combinatorial characterizations of separation systems of graphs and sets, as well as characterizing those separation systems that come from bipartitions of a set – an important special case of set separations. Additionally, we give examples of separation systems which are fundamentally different from separation systems of sets or graphs.
The structure of this paper is as follows: in Section 2, we introduce the terms and notation for separation systems used throughout the paper, and make precise what it should mean that a given separation system has the form of set separations. Our main results, characterizing separation systems and universes consisting of separations of a set or bipartitions of a set, are given in Section 3. Finally, in Section 4, we treat the case of separations of graphs.
2 Separation systems
This paper assumes familiarity with  and uses the same terms and basic notation. In addition, we shall be using the following terms.
For a separation system we write for the set of small separations of : those with . We call a separation co-small if its inverse is small, that is, if .
For a set we write for the separation system of all separations of the set , as defined in : the separation system consisting of all (unoriented) separations of the form , where and are subsets of with , with orientations and , where for oriented separations if and only if and . We write for the universe of all separations of the set : the separation system together with a pairwise supremum and infimum given by
for oriented separations .
Furthermore for a set we let be the separation system of bipartitions of the set : the sub-system of consisting of all those separations with . Similarly, we write for the universe of bipartitions of the set : the sub-universe of containing all separations with and disjoint. Note that we do not insist that and be non-empty, so for all sets .
A map is a homomorphism of separation systems and if it commutes with the involutions of and and is order-preserving. Formally, commutes with the involutions of and if for all . The map is order-preserving if whenever for all . An isomorphism of separation systems and is a bijective homomorphism whose inverse is also a homomorphism. Two separation systems are isomorphic if there is an isomorphism .
Similarly, a map is a homomorphism of universes and if it commutes with the involutions, joins and meets of and . The map commutes with the joins and meets of and if and . An isomorphism of universes, then, is a bijective homomorphism of universes whose inverse is also a homomorphism. Two universes are isomorphic if there is an isomorphism . Clearly, every isomorphism of universes is also an isomorphism of separation systems.
With the above terms and notation we are now able to formally define what it shall mean that a separation system can be implemented by set separations. Given a separation system , we say that can be implemented by set separations if there are a set and a sub-system of such that and are isomorphic. Similarly, we say that can be implemented by bipartitions (of a set) if there are a set and a sub-system of such that and are isomorphic.
If a separation system can be implemented by set separations or by bipartitions, we call both and the isomorphism witnessing this an implementation of by set separations or by bipartitions, respectively.
Finally, for a universe , we say that can be strongly implemented by set separations if there are a set and a sub-universe of such that and are isomorphic universes. Similarly, we say that can be strongly implemented by bipartitions if there are a set and a sub-universe of such that and are isomorphic.
If a universe can be strongly implemented by set separations or by bipartitions, we call both and the isomorphism witnessing this a strong implementation of by set separations or by bipartitions, respectively.
Note that, to show that a separation system can be implemented by set separations or by bipartitions, it suffices to find a ground-set and an injective homomorphism from to or to which is an isomorphism between and its image . In Section 3, most of the proofs will take this approach.
3 Set separations and bipartitions of sets
In this section we shall characterize those separation systems that can be implemented by separations of sets or by bipartitions of sets. We start with a simple observation regarding the shape of small separations in set separation systems:
For any set , the small separations in and are those of the form .
Such separations are clearly small, since . On the other hand, if is small then we have and so . But this implies . ∎
By Lemma 3.1 the small separations in a separation system of sets with ground-set have the following property: for every pair we have . We will show below that this property characterizes separation systems of sets, so let us make it formal:
A separation system is scrupulous if for every pair of small separations we have .
Using the above observation we can characterize the set separation systems as follows:
A separation system can be implemented by set separations if and only if it is scrupulous.
First we check that is scrupulous for any set , from which it follows that any subsystem is scrupulous and so that any which can be implemented by set separations is scrupulous. Let and be small separations of . Then and , so .
Now suppose that is scrupulous. Let be the set of all non-co-small elements of . For any let
There can’t be any , since then we would have both and , so that , which contradicts the fact that isn’t co-small. Thus for any . We shall show that is an implementation of by set separations. It is clear from the definition that is a homomorphism of separation systems, so it remains to check that it is an isomorphism onto its image. That is, we must show that implies that .
So suppose that , that is, and . As we have . If is not small then and it follows that . Similarly, and thus , so if is not small then and hence . But we also have in the remaining case that and are both small, because is scrupulous. ∎
Not every separation system is scrupulous, as the next example shows:
Let be the separation system consisting of the separations and , with the relations as well as and no further (non-reflexive) relations. Then and are small separations with , so is not scrupulous and hence cannot be implemented by set separations.
Example 3.3 demonstrates how any separation system can be modified so as to not have an implementation by set separations: given a scrupulous separation system , one can make this system non-scrupulous by adding a copy of from Example 3.3 to , where separations from are incomparable to those from the copy of . The resulting larger separation system will be non-scrupulous and hence have no implementation by set separations.
However, modifying universes of separations to make them non-scrupulous is not as straightforward as for separation systems due to the existence of joins and meets of any two separations. For universes, being scrupulous is equivalent to another condition on the structure of the small separations:
Let be a universe. Then the following are equivalent:
is scrupulous, i.e. for all small ;
is small for all small ;
is co-small for all co-small .
To see that (i) implies (ii), let be two small separations with and . As is small we have , so . Similarly we have by assumption and hence . But this implies and hence (ii).
To see that, conversely, (ii) implies (i), let be two small separations for which is small. Then
Finally, for the equivalence of (ii) and (iii), note that for all we have by De Morgan’s law, which immediately implies the desired equivalence. ∎
To prove a characterization of universes which can be strongly implemented by set separations we shall need the following technical lemma, which is more about lattices than about separation systems:
Let be a distributive lattice and let and be elements of with and . Then .
By elementary calculations we have
We are now ready to prove an analogue of Theorem 3.2 for universes. As every strong implementation of a universe by set separations is also an implementation of , viewed as a separation system without joins and meets, every universe which can be strongly implemented using separations of sets must be scrupulous by Theorem 3.2. However, being scrupulous is not a sufficient condition for a universe to have a strong implementation by set separations: for every set , the universe as well as all sub-universes of it are (easily seen to be) distributive since intersections and unions of sets are distributive.
So, given a distributive and scrupulous universe , how can we find a strong implementation of ? Let us first suppose that already is a sub-universe of for some set , and see whether we can describe and each just in terms of itself, without making use of .
To this end, for each let be the set of all with , and let be the set of all those . Then we can write any separation as
Thus we can define a map as
and it is easy to check that the map is an isomorphism of universes between and its image in . Thus, the ground-set we defined can be used to obtain a strong implementation of by separations of sets.
In order to mimic this approach in the general case where we do not know already that is a sub-universe of some , we need to find a collection of sets where the sets behave similarly to the sets above. To do this, we shall find some combinatorial properties of the sets , and then take as the set of all which have those combinatorial properties.
In the scenario above where is a sub-universe of some , the first notable property of a set for some is that is up-closed: if and , then and , hence and . Furthermore is closed under taking meets: if , then and , so and hence
Similarly, we get that the complement of in is down-closed and closed under taking joins. Finally, we can say something about the relationship between and the small separations of : namely, that contains the inverse of every small separation of . This is because the small separations of have the form , so for all of them.
By taking as the set of all which have the five properties from the last paragraph, we can prove Theorem 3.6:
A universe of separations can be strongly implemented by set separations if and only if it is distributive and scrupulous.
If has a strong implementation then it is scrupulous by Lemma 3.2 and distributive because is distributive for every .
Now suppose that is distributive and scrupulous. Let be the set of all such that
is up-closed in and closed under taking meets;
is down-closed in and closed under taking joins;
contains all co-small elements of .
For any we have , so is co-small for all . Given and we thus have . Therefore must contain at least one of and , as we cannot have by .
For any let and . By the above argument we have , so takes its image in . This map clearly commutes with the involution, and by and it also commutes with and . It remains to show that is injective. For this let with be given; we shall show that . By switching their roles if necessary we may assume that .
Claim 1: If there is no co-small such that then .
To see this, we wish to find a pair of disjoint subsets of such that
and is up-closed in and closed under taking meets;
and is down-closed in and closed under taking joins;
contains all co-small elements of .
Call such a pair good. We will show later that a maximal good pair will then have and hence witness that ; let us show first that some good pair exists. To see this, let
and . Then satisfies (I) by Lemma 3.4 and (III) by construction, and clearly satisfies (II). Thus is a good pair provided and are disjoint. Suppose they are not disjoint; then and hence there is some co-small with , which contradicts the premise of Claim 1.
Now let be an inclusion-wise maximal good pair, which exists by Zorn’s Lemma. We wish to show as that would imply and in particular . Suppose there exists . By the maximality of there are and with , and by the maximality of there are and with . Set and . Then and with and . Lemma 3.5 now implies , but this contradicts the fact that is up-closed and disjoint from . Therefore and , which proves the claim.
Claim 2: If there is no co-small such that then .
As is equivalent to Claim 2 follows in exactly the same way as Claim 1.
Claim 3: There cannot be co-small such that
To see this, suppose are as in the claim. Let ; this is a co-small separation by Lemma 3.4. We then have and . Applying the involution to the latter inequality gives . Therefore we get that as well as , which by Lemma 3.5 contradicts the assumption that . This proves the claim.
From Claim 3 it follows that the assumption of at least one of Claim 1 or Claim 2 must be satisfied, and hence , which completes the proof. ∎
The next example shows that the assumption of distributivity in Theorem 3.6 is indeed necessary, as there are abstract universes of separations which are not distributive:
Let be an arbitrary non-distributive lattice. Let be the separation system consisting of an unoriented separation for each , with the following relations: and if and only if in , and additionally for all . As is a lattice every pair of separations in has a meet and a join, so is a universe. As is non-distributive, is non-distributive by construction, too. Therefore there can be no strong implementation of by set separations. Furthermore is scrupulous, showing that the distributivity cannot be omitted from Theorem 3.8.
Let us now turn to the topic of (strong) implementations by bipartitions. From Lemma 3.1 it follows that the only small separation of is . This separation is not only small, it is also the least element of . So let us call a separation system fastidious if we have for all small and all . Clearly every fastidious separation system has at most one small separation, and every separation system with an implementation by bipartitions of sets must be fastidious. Furthermore, every fastidious separation system is scrupulous.
Somewhat surprisingly, Theorem 3.6 directly implies that every distributive and fastidious universe has a strong implementation by bipartitions of sets:
A universe of separations can be strongly implemented by bipartitions of sets if and only if it is distributive and fastidious.
Since the only small bipartition of a set is , any separation system implemented by bipartitions is fastidious. Now suppose that is a distributive and fastidious universe. Then is scrupulous, so by Theorem 3.6 there is a strong implementation of by set separations, where is a sub-universe of for some set . As is co-small for any there exists a co-small separation . Since is fastidious this must be the greatest element of . In particular is the unique co-small element of and we have for all . By Lemma 3.1, the image of under is of the form for some . Given any and