Transit Functions and Pyramid-Like Binary Clustering Systems

Binary clustering systems are closely related to monotone transit functions. An interesting class are pyramidal transit functions defined by the fact that their transit sets form an interval hypergraph. We investigate here properties of transit function $R$, such as union-closure, that are sufficient to ensure that $R$ is at least weakly pyramidal. Necessary conditions for pyramidal transit functions are derived from the five forbidden configurations in Tucker's characterization of interval hypergraphs. The first corresponds to $\beta$-acyclicity, also known as total balancedness, for which we obtain three alternative characterizations. For monotonous transit functions, the last forbidden configuration becomes redundant, leaving us with characterization of pyramidal transit functions in terms of four additional conditions.


Introduction
Transit functions have been introduced as a unifying approach for results and ideas on intervals, convexities, and betweenness in graphs and posets [26].Initially, they were introduced to capture an abstract notion of "betweenness", i.e., an element x is considered to be "between" u and v, if x ∈ R(u, v).Monotone transit functions, satisfying R(x, y) ⊆ R(u, v) for all x, y ∈ R(u, v), play an important role in the theory of convexities on graphs and other set systems [31].They also have an alternative interpretation as binary clustering systems [4].Here every cluster is "spanned" by a pair of points in the following sense: if C is a cluster, then there are two points x and y, such that C is the unique inclusion-minimal cluster containing x and y.Binary clustering systems include not only hierarchies but also important clustering models with overlaps, such as paired hierarchies [7], pyramids [19,8], and weak hierarchies [3].
This connection between transit functions and clustering systems suggests investigating "natural" properties of transit function as motivation for properties of clustering systems.Hierarchies and paired hierarchies are frequently too restrictive.On the other hand, weak hierarchies may already be too general e.g. when considering clustering systems of phylogenetic networks [24].Pyramids, i.e., clustering systems whose elements can be seen as intervals, form a possible middle ground.These set systems are equivalent to interval hypergraphs [30,29,27,21].A potential shortcoming of pyramids is that the associated total order of the points may not always have a simple interpretation and that these set systems do not have a compact characterization without (implicit) reference to an ordering of the points.It is of interest, therefore, to consider binary clustering systems that are either mild generalizations or restrictions of pyramids.Naturally, the existence of total order in an interval hypergraph closely liked to notions of acyclicity in hypergraphs, which were originally studied in the context of database schemes [23,2].Here, we connect these concepts with corresponding properties of transit functions and binary clustering systems.
This contribution is organized as follows.In Section 2, we provide the technical background and connect properties of transit functions with the literature on cycle types and acyclicity in hypergraphs.It is proved in [17] that the union closed binary clustering systems are pyramidal.Section 3 considers union-closed set systems and derives an alternative characterization of their canonical transit functions, thereby solving an open problem of [17].In section 4, we consider the weak pyramidality property and a relaxed variant termed (i) that is closely related to axioms studied in earlier work [17].We prove that weakly pyramidal set systems and weak hierarchies satisfying (I) are equivalent and that the corresponding axiom (i) for transit functions is in between (wp) and (o').In section 5, we consider sufficient conditions for weakly pyramidal transit functions.Here we consider two clustering systems between paired hierarchies and weakly pyramidal set systems that are described by the axioms (L1) and (N3O), and study their identifying transit functions.We introduce axiom (l2) and prove that a monotone transit function satisfying both (l1) and (l2) is pyramidal.
Then we turn to the necessary conditions for pyramidal transit functions: Section 6 is concerned with more general, so-called totally balanced transit functions, whose set systems are β-acyclic hypergraphs [1,25] and thus do not contain the first forbidden configuration in Tucker's [30] characterization of interval hypergraphs.We derive three alternative characterizations for this property for totally balanced monotone transit functions.We then proceed in Sections 7 and 8 to analyze the forbidden configurations in Tucker's [30] characterization and finally arrive at a characterization of pyramidal transit functions.We close this contribution in section 9 with a summary of the relationships among the properties considered here and some open questions.

Notation and Preliminaries
Throughout, V is a non-empty finite set and C ⊆ 2 V is a set of subsets of V that does not contain the empty set.Two sets A, B ⊆ V overlap, in symbols A B, if A ∩ B, A \ B and B \ A are non-empty.
Transit Functions and T-Systems Formally, a transit function [26] on a non-empty set V is a function R : V × V → 2 V satisfying the three axioms (t1) u ∈ R(u, v) for all u, v ∈ V .(t2) R(u, v) = R(v, u) for all u, v ∈ V .(t3) R(u, u) = {u} for all u ∈ V .
The transit functions appearing in the context of clustering systems [4] in addition satisfy the monotonicity axiom (m) p, q ∈ R(u, v) implies R(p, q) ⊆ R(u, v).
Transit functions are sometimes called "Boolean dissimilarities".The set systems corresponding to monotone transit functions are slightly more general than binary clustering systems.A characterization was given in [15]: Definition 2.1.A T-system is a system of non-empty sets C ⊂ 2 V satisfying the three axioms (KS) {x} ∈ C for all x ∈ V .(KR) For every C ∈ C there are points p, q ∈ C such that p, q ∈ C implies C ⊆ C for all C ∈ C. (KC) For any two p, q ∈ V holds {C ∈ C|p, q ∈ C} ∈ C.
We say that a set system C is identified by the transit function R if C = {R(x, y)|x, y ∈ V }.
Proposition 2.1.[15] There is a bijection between monotone transit functions R : That is, a set system C is identified by a transit function R if and only if C is a T-system.In this case, R = R C and we call R C the canonical transit function of the set system C, and C R the collection of transit sets of R. A set system is binary in the sense of [4] if and only if it satisfies (KC) and (KR).Axiom (KS) corresponds to (t3), i.e., the fact that all singletons are transit sets.Moreover, a T-system is a binary clustering system if it satisfies As shown in [4,15], binary clustering systems are identified by monotone transit function satisfying the additional condition

Conformal Hypergraphs
The system of transit sets C R has a natural interpretation as the edge set of the hypergraph (V, C R ).The primal graph or two-section H [2] of (V, C R ) is the graph with vertex set V and an edge {x, y} ∈ E(H [2] ) whenever there is C ∈ C R with {x, y} ⊆ C. Axiom (t1) and the fact that R is defined on V × V implies that the primal graph H [2] of (V, C R ) is the complete graph on V .A hypergraph is said to be conformal [28] if every clique of its two-section is a hyperedge or, equivalently, covered by a hyperedge.Therefore we have Observation 2.2.Let R be a monotone transit function.Then (V, C R ) is a conformal hypergraph if and only if R satisfies (a').
Since complete graphs are chordal, and a hypergraph is α-acylic if and only if it is conformal and its two-section is chordal [5], we obtain immediately: ) is a convexity if it satisfies (K1) and (K2).We remark that convexities are usually defined to include the empty set and thus without the restriction of (K2) to A ∩ B = ∅.Here, we insist on ∅ / ∈ C in order for grounded convexities, i.e., those satisfying (KS), to be the same as closed clustering systems.Note that (K2) implies (KC), while the converse is not true.In the language of monotone transit functions, (K2) is equivalent to Axiom (k) was introduced as (m') in [16,15] and renamed (k) in later work to emphasize that it is unrelated to the monotonicity axiom (m).
Weak Hierarchies A clustering system C is a weak hierarchy [3] if for any three sets The family of transit sets of the canonical transit function of a weak hierarchy is a convexity, and hence closed [15].The corresponding canonical transit function satisfies (w) For all x, y, z ∈ V holds z ∈ R(x, y) or y ∈ R(x, z) or x ∈ R(y, z).
Definition 2.2.A clustering system (V, C) is pre-pyramidal if there exists a total order < on V such that for every C ∈ C and all x, y ∈ C we have x < u < y implies u ∈ C. That is, all clusters C ∈ C are intervals w.r.t.<.
As shown in [30,29,21], pre-pyramidal set systems are characterized by the following (infinite) series of forbidden induced sub-hypergraphs: (3) Note that the forbidden configurations are to be interpreted as restrictions of a hypergraph to a subset of its vertices.Sets shown as disjoint in the diagrams therefore may intersect in additional vertices.Furthermore, the number of "small" overlapping sets located completely inside the large ellipse in the fourth case and those located completely inside the intersection of the large ellipses in the fifth case may be any k ≥ 0. It is well known that pre-pyramids are a proper subclass of weak hierarchies, see e.g.[9].
Pre-pyramidal set systems are also known as interval hypergraphs [20].For any three distinct points x, y, z ∈ V , we say that y lies between x and z if every hyperpath connecting x and z has an edge containing y.A hypergraph (V, C) is an interval hypergraph if and only if every set {x, y, z} ⊆ V of three distinct vertices contains a vertex that lies between the other two [20].A clustering system (V, C) is pyramidal if it is pre-pyramidal and closed.Since pre-pyramidal clustering systems are a special case of weak hierarchies [8], we observe that for every monotone transit function with a pre-pyramidal system of transit sets, C R is closed and thus pyramidal.
Definition 2.3.A monotone transit function R is pyramidal if its set of transit sets C R form a pyramidal clustering system.
In other words, R is pyradmidal if there is a total order < on V such that R(u, v) is an interval for all u, v ∈ V .It is worth noting that, despite Duchet's betweenness-based characterization of interval hypergraphs [20], neither of the two classical betweenness axiom [26] need to be satisfied.The transit function R on V = {u, v, x, y} comprising the singletons, R(v, y) = {v, y} and R(p, q) = V otherwise, is monotone and violates (b3).However, it is pyramidal with order x < u < y < v.The "indiscrete transit function", whose transit sets are only the singletons and V , is monotone, pyramidal, and violates (b4).In fact, the pyramidal transit function satisfying (b4) is uniquely defined (up to isomorphism).
Given a total order < on V , we write [u, v] := {w ∈ V |u ≤ w ≤ v} for the intervals w.r.t. the order <.A monotone pyramidal transit function R by construction satisfies [u, v] ⊆ R(u, v) for all u, v ∈ V .Proposition 2.4.Let R be a monotone, pyramidal transit function satisfying (b3) or (b4).Then R(u, v) = [u, v] for all u, v ∈ V .
Proof.Let R be monotone and pyramidal.In particular, R satisfies (a'), i.e., there is p < q such that R(p, q) = [p, q] = V .First suppose R satisfies (b3).If x = q, then q / ∈ R(p, x) since otherwise (b3) implies x ∈ R(q, q) = {q}, contradicting (t3).If x is the predecessor of q w.r.t.<, then R(p, x) = [p, x].Repeating the argument for the predecessor x ∈ R(p, x) = [p, x] of x yields R(p, x ) = [p, x ], and eventually R(p, u) = [p, u] for all u ∈ V .We argue analogously that for p ∈ [p, x] with p = p we have p / ∈ R(p , x) ⊆ [p, x].For the successor w of p, therefore, we we obtain R(w, x) = [w, x].Again, repeating the argument for w ∈ [w, x] we eventually obtain R(u, x) = [u, x] for all u ≤ x.Now suppose R satisfies (b4).Assume that there is x ∈ V such that w ∈ R(p, x) \ [p, x] and thus w ∈ R(p, x) ∩ [x, q] ⊆ R(p, x) ∩ R(x, q) = {x}, contradicting (b4).Thus R(p, x) = [p, x] for all x ∈ V .Similarly, we obtain R(x, q) = [x, q] for all x ∈ V .Now consider x ∈ R(p, x) and suppose R(x , x) = [x , x], i.e., there is w Totally balanced clustering systems The first forbidden configuration in Eq.( 3) amounts to the absence of a so-called weak β-cycle [23,18], which is defined as a sequence of n ≥ 3 sets C 1 , . . ., C n and vertices x 1 , . . ., x n such that, for all i, x i ∈ C i ∩ C i+1 and x i / ∈ C k for any k / ∈ {i, i + 1} (where indices are taken modulo n).Hypergraphs without a weak β-cycle are known as β-acyclic or totally balanced, see e.g.[1,25].Since a β-cycle of length 3 amounts to ∈ C i+2 , we note that C is a weak hierarchy if and only if it does not contain a β-cycle of length 3 [3].From the point of view of clustering systems, totally balanced (β-acyclic) hypergraphs have been studied in [13].
Paired Hierarchies A set system C is a paired hierarchy if every cluster C ∈ C overlaps at most one other cluster C ∈ C [7].A characterization of paired hierarchies in terms of their transit functions can be found in [9,15].Hierarchies A set system C on V is a hierarchy if V ∈ C, all singletons belong to C and A ∩ B ∈ {A, B, ∅} for all A, B ∈ C. Several alternative characterizations of monotone transit functions whose transit sets form a hierarchy are discussed in [16], see also [9].

Acyclicity in Hypergraphs
There is extensive literature concerned with notions of acyclicity in hypergraphs [23,2,22,10].A stronger condition than β-acyclicy is γ-acyclicity, which can be phrased as follows: A hypergraph is γ-acyclic if it contains neither a pure cycle nor a so-called γ-triangle.
If C is not a weak hierarchy, then there are three pairwisely overlapping sets [15], and thus any pair of overlapping clusters C 1 C 2 together with V forms a γ-triangle.Thus a γ-acyclic weak hierarchy contains no overlapping clusters and thus are hierarchies.Conversely, hierarchies are γ-acyclic since every pure cycle consists of a sequence of consecutive overlapping clusters, and every γ-triangle contains two overlapping sets.Therefore, we have Proposition 2.5.Let R be a monotone transit function.Then C R is γ-acyclic if and only if C R is a hierarchy.
3 Union-Closed Set Systems A set system (V, C) is union-closed if it contains all singletons and satisfies In [17] we considered the following two properties: A monotone transit function satisfies (uc) if and only if the corresponding T-system satisfies (UC).Axiom (u) appeared as a property of cut-vertex transit functions of hypergraphs in [14] and as property (h") in [16] in the context of characterizing hierarchical clustering systems.It was studied further in [17], where it was left as an open question whether weak hierarchies with canonical transit functions that satisfy (u) are union-closed.The following result gives an affirmative answer: Proof.Suppose R satisfies (uc).Then [17,Thm.3]shows that R satisfies (w), and [17, Lemma 8] establishes that R satisfies (u).
For the converse, assume that R satisfies (u) and (w) and suppose that R(x, y) ∩ R(u, v) = ∅ and there exists p By assumption, there exists ).The case is impossible since p ∈ R(a, q) implies p ∈ R(u, v) contradicting our assumptions; thus p ∈ R(a, d 1 ).By contraposition, p / ∈ R(a, d 1 ) implies p / ∈ R(d 1 , q); in this case p, q, and d 1 violate (w).Therefore p ∈ R(d 1 , a) and thus R(p, a) ⊂ R(d 1 , a) by monotonicity.This implies and thus d 1 , d 2 , and q violate (w).Therefore, we must have q ∈ R(a, d 2 ) and hence Otherwise, there exists . Using the same arguments, there exists a larger Fig. 4 in [17] shows that the converse is not true.Thm.3.1 and Prop.3.2 together answer affirmatively the questions in [17] whether (u) and (w) together are sufficient to imply that a monotone transit function R is pyramidal.
Paired hierarchies and union-closed binary clustering systems are proper sub-classes of pyramidal clustering systems.There are, however, binary clustering systems that are union-closed but not paired hierarchies and clustering systems that are paired hierarchies but not union-closed:

Weakly Pyramidal Transit Functions
Ref. [27] characterizes pre-pyramidal set systems with {A, B, C} as those that are weak hierarchies and satisfy (WP) If A, B, C have pairwise non-empty intersections, then one set is contained in the union of the two others.
For larger set systems C the condition is still necessary, but no longer sufficient.Thus the term weak pre-pyramids has been suggested for weak hierarchies that satisfy (WP).The axiom can be translated trivially to the language of transit functions: The third forbidden configuration in Eq.( 3) suggests considering set systems satisfying the following property: Let us now consider the following related property for transit functions: Observation 4.1.Let R be a monotone transit function, and C R the corresponding set of transit sets.Then C R satisfies (I) if and only if R satisfies (i).
Similarly, in the second case, we obtain B ⊆ C, and thus (I) holds.
Lemma 4.3.Let R be a monotone transit function satisfying (wp), then R satisfies (i).
Proof.Consider ∅ = R(x, y) ∩ R(u, v) ⊆ R(p, q) where R(p, q) \ (R(x, y) ∪ R(u, v)) = ∅ and assume, for contradiction, that neither R(x, y) ⊆ R(p, q) nor R(u, v) ⊆ R(p, q) is true.Then there exists w 1 ∈ R(x, y) \ R(p, q), and Hence none of the sets R(x, y), R(u, v), and R(p, q) is contained in the union of the other two, contradicting (wp).
The transit function R and sets system C R in Example 4.4 below shows that the converses of both Lemma 4.2 and 4.3 do not hold.d, f } and all other sets are singletons or V .Here R is monotone and satisfies (i) but the sets {a, b, c}, {a, d, e}, {c, d, f } violate (wp).
Moreover, axioms (i) and (w) are independent.The canonical transit function in Fig. 1B satisfies (i) but violates (w) and in Fig. 1C, it satisfies (w) but violates (i).A transit function property that is weaker than the axiom (u) and proved to be weaker than the axiom (wp) in [17] is: We next show that (i) is in general weaker than (wp) and stronger than (o'): Lemma 4.7.If a monotone transit function R satisfies (i), then it also satisfies (o').
Proof.Let R be a monotone transit function satisfying (i) and suppose, for contradiction, that R violates (o').Then there exist u, v, z In this case, we consider a new point u k 2 ∈ R(u, v) that is not in any of the three sets.Such a point exists since R violates (o').If R(u i , z) ⊂ R(u k 1 , z) holds, then consider the sets R(u j , z), R(u k 1 , z), and R(u k 2 , z).If at least one of the three sets is contained in another, we consider a new point u k 3 ∈ R(u, v) which is not in any of the three sets.Again, such a point exists since R violates (o').Continuing in this manner, we obtain an infinite number of points u k 1 , u k 2 , . . .because in each step, a new point from R(u, v) is added.However, this contradicts the fact that R(u, v) is finite.After a finite number of steps, we, therefore, encounter Case 2: R(u i , z), R(u j , z), R(u k , z) overlap pairwisely.Suppose the intersection of two sets is contained in the third, say, ) contradicting the assumption that the three sets overlap pairwisely.Therefore we have ).Now consider the sets R(v 1 , z), R(v 2 , z), and R(v 3 , z).Since the sets R(u i , z), R(u j , z), and R(u k , z) overlap pairwisely, by (m), the sets R(v 1 , z), R(v 2 , z), and R(v 3 , z) also overlap pairwisely.If the intersection of two sets is contained in the third one, then (i) again implies that one of the sets is contained in another, contradicting the assumption that the three sets overlap pairwisely.Then there exist points Repeating these arguments, we eventually obtain points Example 4.8 below shows that the converse need not be true: d} and all other sets are singletons or V .R is monotone, satisfies (o') but not (i).

Between paired hierarchies and pyramids
Consider a set system C satisfying the following properties: Axiom (N3O) appeared in recent work on the clustering systems of so-called galled trees, a special class of level-1 phylogenetic networks [24].By definition, a paired hierarchy trivially satisfies (N3O) and (L1) since, in this case, we must have A = C.Moreover, these axioms are related as follows: By symmetry, the same is true is A does not overlap B and B C. Finally, suppose A B and B C. According to (L1), three cases may occur.In case (i), We can translate (L1) and (N3O) in terms of transit functions as follows: Corollary 5.6.If a monotone transit function R satisfies (l1), then R satisfies (n3o).
Proof.Suppose R does not satisfy (w).Then there exist points x, y, z ∈ V such that x / ∈ R(y, z), y / ∈ R(x, z), and z / ∈ R(x, y), and thus the three transit sets overlap pairwisely, i.e., we have R(x, y) R(y, z), R(y, z) R(z, x), and R(z, x) R(x, y), violating (n3o).Now suppose R satisfies (n3o) and consider three transit sets A, B, C ∈ C R with pairwise nonempty intersections.If at least one set is contained in another set, then R satisfies (wp).Otherwise, A B, B C, and A C contradict (n3o).
Nevertheless, (n3o) is not sufficient to guarantee that R is pyramidal, which is clear from Fig. 1A.Moreover, Example 5.8 shows that the converse of Lemma 5.7 is also not true.A monotone transit function R satisfying (l1) satisfies (w) and, therefore, also (k2).Consequently, we have Observation 5.9.If the monotone transit function R satisfies (l1), then C R is closed.
Let us now consider the following axiom, which is related to, but weaker than, the union-closure condition (uc): If R is a monotone transit function such that C R is a paired hierarchy, then R satisfies (l2).In the language of set systems, (l2) clearly implies the following property: However, the following Example 5.10 shows that (L2') is not the correct "translation" of (l2): Condition (l2) on transit functions is more restrictive than (L2') since, in addition to the behavior of the transit sets, it also requires the existence of two "reference points" s and t that satisfy an additional condition.An axiom for R corresponding to (L2') would require only the existence of s and t such that R(x, y) ∪ R(p, q) ∪ R(u, v) = R(s, t).A related property for general set systems is : Lemma 5.11.If R is a monotone transit function satisfying (w) and (l2), then C R is a weak hierarchy satisfying (L2') and (L2").
Proof.Suppose R satisfies (w) and (l2).Then C R is obviously a weak hierarchy satisfying (L2').Set A = R(x, y), B = R(p, q), and C = R(u, v), i.e., A, B, C ∈ C R and assume A B C and Then by (l2), there exist s ∈ R(x, y) \ R(p, q) and t ∈ R(x, y)\R(p, q) such that R(s, t) = A∪B∪C, and by monotonicity, R(s, t) ⊆ D, i.e., A∪B∪C ⊆ D, and thus (L2") is satisfied.
In the following Example 5.12, we see the independence of axioms (l1) and (l2) and their connection with the axioms (w), (wp), and (uc): there is no linear ordering on V compatible with C. Thus (V, C) is weakly (pre-)pyramidal but not (pre-)pyramidal.(B) The set system C comprises the singletons, V , and the three pairs {p, q} with p, q ∈ {a, b, c}.These three pairs intersect pairwisely, but {a, b} ∩ {a, c} ∩ {b, c} = ∅; thus, C is not a weak hierarchy.The canonical transit function satisfies R(p, q) = {p, q} for p, q ∈ {a, b, c} and R(p, d) = V for p ∈ {a, b, c}.(C) The set system C consists of the singletons, the three edges {a, b}, {b, c}, {b, d}, and V .It is a weak hierarchy.Its canonical transit function R satisfies R(p, q) = V if and only p = q and p, q ∈ {a, c, d}.From Examples 5.12 and 5.15, we see that (l1) and (l2) are independent.Examples 5.12 and 5.13 show that (uc) and (l1) are independent.Furthermore, Examples 5.14 and 5.15 show that (l2) is independent of both (w) and (wp).The transit function R in Example 5.10, furthermore, satisfies (w), (wp), and (l1) but violates (l2), and its transit set C R is not pyramidal.
Axiom (l2) can be seen as a relaxation of the union-closure property.Indeed, we have Lemma 5.19.Let R be a monotone transit function satisfying (uc), then R satisfies (l2).
Proof.Since (l1) implies (w), C R cannot contain a 3-cycle.Also, it follows from (l1 In particular, we have Proof.Since (l1) implies that C R satisfies (L1), we conclude from Obs. 5.1 and Lemma 5.2 that C R is closed and satisfies (W) and (WP).Furthermore, (L1) implies that C R cannot contain the second, third, and fourth forbidden configurations of an interval hypergraph in Eq.( 3).Since C is closed, the fifth configuration is also ruled out because the intersection of the large sets in the fifth configuration is again a cluster.Thus, the fifth configuration contains the fourth one as a subhypergraph.A closed clustering system satisfying (L1) is, therefore, either pyramidal or contains a hypercycle for all 1 ≤ i ≤ n (indices taken mod n), for some n > 3.By Lemma 5.20, (l1) and (l2) rule out the existence of such a hypercycle.Axiom (L1) thus enforces the existence of a linear order locally.It is insufficient to ensure global consistency with a linear order, however.
As a consequence of Thm.5.21 and Example 5.17, the transit functions satisfying (l1) and (l2) are a proper subset of the pyramidal transit functions, which are a proper subset of the weakly pyramidal transit functions.On the other hand, monotone transit functions of paired hierarchies satisfy (l1) and (l2).Example 5.13 again shows that the converse is not true.

Totally Balanced Transit Functions
In [16] we considered (u) as a key property of hierarchies.Fig. 4 in [17] shows, however, that pyramidal transit functions do not necessarily satisfy (u).Here we consider a natural generalization: Clearly, (u) implies (u3).Example 3.3 shows that the converse is not true.
The following Example 6.3 shows that the independent axioms (w), (wp), and (u3), even together, do not imply that a monotone transit R function is pyramidal.
and the other sets are singletons or V .Here R is monotone and satisfies (w), (wp), and (u3) but is not pyramidal.
Let < be a linear order on V and consider a subset and [x j , x k ] do not overlap.This simple observation suggests considering the following conditions: It follows immediately from the definition that (hc) implies (tb).Example 5.8 shows that (tb) does not imply (n3o).Fig. 1A shows that (n3o) does not imply (tb).The example in Fig. 1B, furthermore, shows that (hc) does not imply (wp).Moreover, the following Example 6.4 shows that (wp) does not imply (hc).The transit function corresponding to the set system in Fig. 1B trivially satisfies (l2) but violates (tb).The monotone transit function in Example 5.15 is pyramidal and satisfies (tb) but violates (l2).In summary, (tb) is independent of the axioms (n3o), (wp), and (l2).
Example 6.4.Consider the transit function Here R satisfies (wp) but violates (hc).Lemma 6.5.Let R be a monotone transit function on V .R satisfies (tb) if and only if R satisfies (hc).
Proof.It follows immediately from the definition that (hc) implies (tb).To prove the converse, we proceed by induction in n = |V |.First, consider the base case n = 3 and assume that the monotone transit function R on V = {a, b, c} satisfies (tb).W.l.o.g., suppose R(a, b) ⊆ R(a, c).Then by the monotonicity of R, we have R(a, c) = {a, b, c} and thus R(b, c) ⊆ R(a, c), i.e., R satisfies (hc).Now suppose |V | > 3 and the assertion holds for all proper subsets of V .Since R satisfied (tb), there exists a ∈ V such that R(a, u) and R(a, v) do not overlap for any pair u and v. Consider V := V \ {a}.The induction hypothesis stipulates that there exist b, c ∈ V such that for all u, v ∈ V , the sets R(b, u) and R(b, v) do not overlap and the sets R(c, u) and R(c, v) do not overlap.Since R satisfies (tb) we know that R(a, b) and R(a, c) do not overlap.W.l.o.g., we may assume R(a, b) ⊆ R(a, c) and thus b ∈ R(a, c).Monotonicity now implies R(b, c) ⊆ R(a, c).If (hc) does not hold, then there is y ∈ V such that R(a, c) R(c, y), which implies R(b, c) R(c, y) with y ∈ V , a contradiction.Thus R(y, c) ⊆ R(a, c) for all y ∈ V .Since R(a, a) = {a} ⊆ R(a, c), the statement holds for all y ∈ V , and thus R satisfies (hc).
In the following, we prove that (w) is a generalization of (tb).Lemma 6.6.If R is a monotone transit function satisfying (tb) then it also satisfies (w).
Proof.Since R satisfies (tb) for any set of three distinct vertices V = {a, b, c} there exists x ∈ {a, b, c} such that R(x, a ) and R(x, a ) do not overlap for {a , a } = V \ {x} and thus R(x, a ) ⊆ R(x, a ) or R(x, a ) ⊆ R(x, a ).This implies a ∈ R(x, a ) or a ∈ R(x, a ), and thus R satisfies (w).
The converse is not true, however.The example in Fig. 1A satisfies (w) but violates (tb) and thus also (hc).Next we prove that the properties (hc) and (tb) are satisfied by every pyramidal transit functions.Theorem 6.7.Let R be a pyramidal transit function, then R satisfies (hc).
Proof.Let R be a pyramidal transit function on V .Then there exists a linear order < on V such that R(x, y) is an interval with respect to < for every x, y ∈ V .Let V ⊆ V and let x = min{x ∈ V }, y = max{x ∈ V }.Let z, z ∈ V and z ≤ z .Then x ≤ z ≤ z ≤ y.Now, axioms (t1) and (t2) implies that x, z ∈ R(x, z ).Therefore, [x, z ] ⊆ R(x, z ), which implies z ∈ R(x, z ).Thus, z, y) and (m) implies R(z , y) ⊆ R(z, y).In summary, R satisfies (hc).
The converse is not true, however.The Example 6.3 gives a monotone transit function satisfying (hc) that is not pyramidal.Even though it also satisfies (m), (hc), (w), (wp) and (u3), it is not pyramidal because C R contains second forbidden configuration in Eq.( 3).
In the following theorem, we prove that axiom (tb) characterizes the canonical transit function of a totally balanced clustering system: Lemma 6.8.Let R be a monotone transit function.Then R satisfies (tb) if and only if (V, C R ) is totally balanced.
Proof.Let R be a monotone transit function satisfying (tb).Suppose that (C 1 , . . ., C n ) is a weak βcycle in (V, C R ).Consider x i ∈ C i ∩C i+1 for i = 1, . . ., n−1 and Consider the set V = {x 1 , . . ., x n } ⊆ V .For each x i , therefore, there exist x j , x k ∈ V such that R(x j , x i ) R(x i , x k ), thus the set V = {x 1 , . . ., x n } violates (tb), a contradiction.
We note that Lemma 6.8 is similar to Prop.3 of [12], which states that a hypergraph H is totally balanced if and only if it admits a so-called totally balanced ordering.Lemma 6.9.If R is a monotone transit function satisfying (tb), then R satisfies (u3).
Proof.Lemma 6.8 and Lemma 6.6 imply that R satisfies (w) and thus C R cannot contain first forbidden configuration.Using the same arguments as in the proof of Lemma 6.1, we conclude that R satisfies (u3).Example 6.2 shows that the converse is not true.The definition of weak β-cycles and the axiom (w) suggest to consider the following axiom for a transit function: ) for all k (indices taken modulo n), then there exists some j with 1 ≤ j ≤ n such that v j ∈ R(v i , v i+1 ) for some i / ∈ {j, j − 1}.
For n = 3, (tb') reduced to (w).A monotone transit function satisfying (tb') thus in particular satisfies (w).Lemma 6.10.Let R be a monotone transit function.Then R satisfies (tb') if and only if (V, C R ) is totally balanced.
A useful property of totally balanced and monotone transit functions is the following: (tb2) For all ∅ = W ⊆ V there exist x, y ∈ W such that W ⊆ R(x, y).Lemma 6.11.Let R be a monotone transit function.If R satisfies (tb) R satisfies (tb2).
Proof.If W = {x} or W = {x, y} then W ⊆ R(x, y) by (t1).It therefore suffices to consider |W | ≥ 3. First assume that R satisfies (tb).Thus there exists x ∈ W , such that, for all u, v ∈ W , either R(x, u) ⊆ R(x, v) or R(x, v) ⊆ R(x, u).Thus there is w ∈ W such R(x, u) ⊆ R(x, w), and thus u ∈ R(x, w) for all u ∈ W . Axiom (P2) can be translated into a transit axiom as follows: Corollary 7.3.Every pyramidal transit function satisfies (p2).
Example 7.8 shows that the converse is not true even if R satisfies (tb), (wp), and (p3).

Fourth and Fifth forbidden configurations
Let us now turn to the fourth and fifth forbidden configurations.We first note that if R satisfies (k) in addition, then C R is closed.In this case, a fifth forbidden configuration in Eq.3 also contains a fourth forbidden configuration as a subhypergraph.Therefore, it suffices to rule out the first four forbidden configurations to ensure that a monotonous transit function satisfying (k) is pyramidal.This is in particularly the case for monotonous transit functions satisfying (w).To address the fourth forbidden configuration, we consider the following property: (p4) If u, v, y, x 1 , . . ., x n ∈ V for n ≥ 3 satisfy (i) x i / ∈ R(x j , x j+1 ) for all i and j / ∈ {i − 1, i}, Since (tb) implies (w) and (p4) implies (wp), we see that (w) and (p4) imply (wpy).However, Examples 8.5, 7.4, 7.9, and 6.3 show that (wpy) is independent of each of the conditions (p2), (p3), (p4), and (tb).The example in Fig1A shows that (w) and (p4) together do not imply (tb).
We are now in the position to give a characterization of pyramidal transit functions as a translation of Tucker's characterization of interval hypergraph [30,29,21] to the realm of transit functions.Theorem 8.6.A transit function R satisfies (m), (tb), (p2), (p3), and (p4) if and only if R is pyramidal.
Proof.Let R be a monotone transit function with the set of transit sets C R .If R is pyramidal, then it is, in particular, totally balanced and thus satisfies (tb) by Thm.6.12.Furthermore, R satisfies axioms (p2), (p3), and (p4) by Corollaries 7.3, 7.7, and Lemma 8.2, respectively.For the converse, we assume that (tb), (p2), (p3), and (p4) hold.We argue that this implies that C R cannot contain any of the five forbidden configurations of Tucker's characterization of interval hypergraphs.First, we observe that (tb) implies that C R is totally balanced and thus does not contain the first forbidden configuration, i.e., weak β-cycles.Axioms (p2) and (p3) imply that C R satisfies (P2) and (P3), which by construction exclude the second and third forbidden configuration, respectively.Finally, if R satisfies (p4), then C R does not contain the fourth forbidden configuration.We have already noted above that, since (tb) implies (k), every fifth forbidden configuration contains a fourth forbidden configuration.Thus (tb) and (p4) imply that a fifth forbidden configuration cannot appear in C R .Taken together, C R is pyramidal.The examples discussed above, furthermore, show that (p2), (p3), (p4), and (tb) remain independent of each other for monotone transit functions and thus no subset of these four conditions is sufficient.

Discussion
In this contribution, we have obtained a characterization of the pyramidal transit function based on Tucker's characterization [30] of interval hypergraphs in terms of forbidden configurations.Theorem 8.6 used two first-order axioms (p2) and (p3) as well as the monadic second-order axioms (tb) and (p4).All axioms presented in this paper except (p4), (tb2) and the three equivalent conditions (hc), (tb), and (tb'), are first order axioms.We have proved, therefore, that the transit functions of union closed set systems and weakly pyramidal set systems are first order axiomatizable.Also, paired hierarchies are proved to be first order axiomatizable [9].In the light of Theorem 8.6, we strongly suspect that pyramidal clustering systems will not be first-order axiomatizable.Thus we may note that (py) lies between two pairs of first order axiomatizable clustering systems, namely (PH) and (WPY), and (UC) and (WPY), respectively.
Moreover, we resolved an open question from earlier work [17] by showing that a transit function is union-closed if and only if it satisfies (u) and (w).We introduced a pair of conditions (l1) and (l2) that is sufficient for pyramidal transit functions and elaborated on necessary conditions, particularly the weakly pyramidal property and total-balancedness.The implications among all the axioms discussed in this contribution are summarized in Fig. 3. Some relationships remain unclear.In particular, it remains open whether (l2) and (w) together are sufficient to imply (tb) or (u3).In some cases, furthermore, conditions that seem natural for transit functions do not have obvious "translations" to the more general setting of set systems.For example, we do not have an equivalent for (l2) or (p4) in the language of set systems.One might argue that (p4) looks rather contrived beyond being designed to rule out the fourth forbidden configuration.It would certainly be interesting to know whether it could be replaced by simpler condition with a more direct translation to set systems.A hypergraph is arboreal if there is a tree T such that every hyperedge induces a connected subgraph, i.e., a subtree, of T [6, ch.5.4].An interval hypergraph thus is an arboreal hypergraph for which T is a path.Arboreal hypergraphs have been studied from the point of view of cluster analysis and their corresponding dissimilarities in [11], suggesting that the transit functions associated with binary arboreal clustering system also will be of interest.

Example 3 . 3 .
Consider the monotone transit function R on V = {a, b, c, d} defined by R(a, b) = {a, b}, R(b, c) = {b, c} and other sets are singletons and V .Here C R is a paired hierarchy but not union-closed.

Lemma 4 . 5 .
Suppose C is a weak hierarchy.Then (I) implies (WP).Proof.Suppose A, B, C ∈ C intersect pairwisely.Then A ∩ B ∩ C = ∅ and we may assume, w.l.o.g., A ∩ B ∩ C = A ∩ B, and thus A ∩ B ⊆ C. Then either C \ (A ∪ B) = ∅, i.e., C ⊆ A ∪ B, or (I) implies A ⊆ C and thus also A ⊆ B ∪ C or B ⊆ C and thus also B ⊆ A ∪ C. In either case, therefore, one of the three sets is contained in the union of the other two, and thus C satisfies (WP).

Corollary 4 . 6 .
If C is a weak hierarchy, then (I) and (WP) are equivalent.

Lemma 5 . 1 .
If C satisfies (L1), then C satisfies (WP).Proof.Suppose A, B, C ∈ C pairwisely intersect.Then either A B and B C, or B and at least one of A and C are nested.In the latter case, we may assume w.l.o.g.B ⊆ C or C ⊆ B, and thusB ⊆ A ∪ C or C ⊆ A ∪ B. If A B and B C then (L1) implies (i) A ⊆ C ⊆ B ∪ C, or (ii) C ⊆ A ⊆ A ∪ C, or (iii) B ⊆ A ∪ C.In either case, one of A, B, C is contained in the union of the other two sets.Lemma 5.2.Let C be a set system satisfying (L1).Then C is a weak hierarchy.Proof.Let A, B, C ∈ C. First, suppose A, B, and C do not overlap.Then either A, B, C are pairwise disjoint, in which case A ∩ B ∩ C = ∅ = A ∩ B or the three sets are nested.Assume, w.l.o.g., that A ⊆ B ⊆ C. Then A ∩ B ∩ C = A ∩ B = A. Second, assume A B and B does not overlap C. The following four situations may occur: (a) B

Corollary 5 . 3 .
If C satisfies (L1), then it is weakly pyramidal.Lemma 5.4.Let C be a set system.Then (L1) implies (N3O).Proof.Suppose axiom (L1) holds, A B and B C, and we have neither A ⊆ C nor C ⊆ A. In case (iii), one of the following situations are possible: (a) A ∩ C = ∅, (b) B = A ∩ C, (c) B = A ∪ C, (d) ∅ = A ∩ C B A ∪ C. In case (b), we have B ⊆ A and B ⊆ C, and thus B does not overlap A and C, a contradiction.In case (c), A ⊆ B and C ⊆ B imply that B does not overlap A and C, again a contradiction.In case (d), we have A C. By (L1), A B and A C imply that B ∩ C ⊆ A ⊆ B ∪ C, and thus A ∩ B ∩ C = B ∩ C, since the other two options of (L1) contradict the assumption that B and C overlap.By the same argument, A C and B C and (L1) implies A ∩ B ∩ C = A ∩ B. As in the proof of Lemma 5.2, A B and B C implies A ∩ B ∩ C = A ∩ C. Thus, in case (iii), we have either A ∩ C = ∅ or A C, in which case A ∩ B = A ∩ C = B ∩ C must hold.Therefore, B ⊆ A ∪ C implies B = B ∩ (A ∪ C) = (A ∩ B) ∪ (C ∩ B) = (B ∩ C), contradicting B C. Therefore, A ∩ C = ∅.The following Example 5.5 shows that the converse is not true.Example 5.5.Let A = {a, b}, B = {b, c, d}, C = {d, e}, V and the singletons be the sets in a set system C on V = {a, b, c, d, e}.Then, C satisfies (N3O) but violates (L1) as B A ∪ C.

Figure 1 :
Figure 1: Three set systems with corresponding canonical transit functions that serve as counterexamples, ruling out potential implications between axioms in this contribution: (A) The set system C on V comprising the singletons, V , and the four edges C 1 = {a, b}, C 2 = {b, c}, C 3 = {c, d} and C 4 = {d, a} is a weak hierarchy (since any triple of sets that intersect pairwise contains either V or a singleton), and satisfies axiom (WP).Since the four edges form a 4-cycle (C 1 , C 2 , C 3 , C 4 ) in (V, C),there is no linear ordering on V compatible with C. Thus (V, C) is weakly (pre-)pyramidal but not (pre-)pyramidal.(B) The set system C comprises the singletons, V , and the three pairs {p, q} with p, q ∈ {a, b, c}.These three pairs intersect pairwisely, but {a, b} ∩ {a, c} ∩ {b, c} = ∅; thus, C is not a weak hierarchy.The canonical transit function satisfies R(p, q) = {p, q} for p, q ∈ {a, b, c} and R(p, d) = V for p ∈ {a, b, c}.(C) The set system C consists of the singletons, the three edges {a, b}, {b, c}, {b, d}, and V .It is a weak hierarchy.Its canonical transit function R satisfies R(p, q) = V if and only p = q and p, q ∈ {a, c, d}.
which is the case we have already ruled out.Hence C R is totally balanced (β-acyclic).Theorem 5.21.If R is a monotone transit function satisfying (l1) and (l2), then C R is pyramidal.

Figure 2 :
Figure 2: A, B, and C are set systems corresponding to the monotone transit functions in Examples 6.3 , 7.4 and 7.8 respectively.

Example 7 . 4 .Lemma 7 . 5 .
Let R on V = {a, b, c, d, e} be defined by R(a, b) = {a, b}, R(b, c) = R(c, d) = {b, c, d}, R(b, d) = {b, d}, R(d, e) = {d, e}, R(a, d) = R(a, e) = R(b, e) = {a, b, d, e} and all other sets are singletons or V .R satisfies (m), (wp), (tb), and (p2).Here R(a, b) R(c, d), R(d, e) R(c, d), R(a, d) R(c, d).That is, C R contains a third forbidden configuration in Eq.3.Therefore, C R is not pyramidal.See Fig.2A.Example 6.3 shows that neither (w) nor (wp) implies (p2).The transit function in Fig. 1C satisfies (p2) but violates (wp).In Example 6.4, R satisfies (p2), but the points c, e, and f violate (w).Hence, (w), (wp), and (p2) are mutually independent.Also, (wpy) and (p2) are independent.Moreover, the transit function R in Example 6.4 violates (tb) as C R contains the pure-cycle (R(c, f ), R(c, e), R(e, f )), and Example 6.3 shows that (tb) does not imply (p2); thus properties (tb) and (p2) are independent.Let us now turn to the third forbidden configuration: (P3) If A B, B C, B D, and A ∪ C ⊆ D, then A ∩ C = ∅.(P3') If A B, B C, A ∩ C = ∅, and A ∪ C ⊆ D, then B ⊆ D. Axioms (P3) and (P3') by design rule out the third forbidden configuration in Eq.3.The two formulations are equivalent.To see this, consider four sets A, B, C, D ⊆ V and assume A B, B C, and A ∪ C ⊆ D. Under these assumptions, we have B ∩ D = ∅ and thus B D is equivalent to B ⊆ D. The statement "B D implies A ∩ C = ∅" thus is equivalent to the contra-positive of "A ∩ C = ∅ implies B ⊆ D".Every pyramidal set system C satisfies (P3).Proof.Let C be pyramidal.First, suppose (P3) does not hold, i.e., there exist four sets A, B, C, D that are intervals such that A B, B C, B D and A ∪ C ⊆ D, but A ∩ C = ∅.Then B is located between A and C, i.e., A ∪ B ∪ C is an interval whose endpoints are located in A \ B and C \ B, respectively, and therefore in D. Using again that C is pyramidal, we obtain A ∪ B ∪ C ⊆ D, contradicting B D. Lemma 7.6.If a set system C satisfies (L1), then it also satisfies (P3).
Proof.Let A, B, C, D ∈ C such that A B, B C, B D, and A ∪ C ⊆ D. Since (L1) holds, A B and B C together implies A ⊆ C, or C ⊆ A, or A ∩ C ⊆ B ⊆ A ∪ C. The third alternative cannot occur because it yields B ⊆ A ∪ C ⊆ D and thus contradicts B D. In the first two cases we have A ∩ C = A or A ∩ C = C, and thus A ∩ C = ∅, i.e., (P3) holds.

Figure 3 :
Figure 3: Summary of implications among properties of monotone transit functions.