Interval decomposition lattices are balanced

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in a linearly ordered set. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions of a set, in a general sense of intervals defined axiomatically. This characterization is used to show that the lattice of interval decompositions is balanced.


Preliminaries
Decompositions into intervals were first studied by Hausdorff [12,13], in the context of linearly ordered sets, then extended to partially ordered sets and graphs (see Sabidussi [20]), appearing in particular in the study of comparability graphs (Gallai [9]). The concept of decomposition was extended to hypergraphs and directed graphs by Dörfler and Imrich [4] and Imrich [3], and to higher arity relational structures by Fraïssé [6,7,8]. A general, abstract theory of decompositions was first presented by Möhring and Radermacher [17] and Möhring [16]. Under mild stipulations about what is to be considered an interval, interval decompositions constitute a complete lattice. In [17] it was proved that this lattice is semimodular whenever it is of a finite length. This result was extended in [5] to arbitrary interval decomposition lattices, and their meet-irreducible elements were also described. Another proof for semimodularity can be found in [14]. We note that in graph theory, interval decompositions are closely related to the transitive orientation problem (see [9] and [15]). In the present paper, we prove further properties of the lattice of interval decompositions. First, we characterize the join-irreducible elements in this lattice, and using this result we show that the lattice is balanced. As a consequence, several other properties of the lattice of interval decompositions are deduced, and the case when this lattice is distributive is characterized.
A closure system pV, Qq, Q Ď PpV q is called algebraic if the union of any nonempty chain of closed sets is closed. An interval system pV, Iq was defined in [5] as an algebraic closure system with the following properties: (I 0 ) txu P I for all x P V and ∅ P I, (I 1 ) A, B P I and A X B ‰ ∅ imply A Y B P I, (I 2 ) or any A, B P I the relations A Ę B and B Ę A imply AzB P I (and BzA P I).
Examples of interval systems given in [5] include modules of graphs and relational intervals. These latter include order intervals in linearly ordered sets. In any closure system pV, Qq, a set A P Q is called a strong set, if for every B P Q, A X B " ∅ or A Ď B or B Ď A. The empty set, V and the singletons tau, a P V are called improper strong sets. Let S stand for the set of strong sets in pV, Qq; then pV, Sq is a closure system satisfying conditions (I 1 ) and (I 2 ) trivially. Let pV, Qq be algebraic and satisfy condition (I 0 ), then any singleton and ∅ are strong sets and pV, Sq is an interval system.
We note that restricting a closure system pV, Qq to a nonempty set A Ď V we obtain again a closure system pA, Q A q with Q A " tQ X A | Q P Qu. Clearly, for any A P Q we have Q A Ď Q, and pA, Q A q is an interval system whenever pV, Qq is an interval system. Definition 1.1. A decomposition in a closure system pV, Qq is a partition π " tA i | i P Iu of the set V such that A i P Q, for all i P I. The decomposition π is said to be proper, if it has at least two distinct blocks A i . If pV, Qq is an interval system, then π is called an interval decomposition. The set of all decompositions in pV, Qq is denoted by DpV, Qq.
Let PartpV q denote the lattice of all partitions of V . Since DpV, Qq Ď PartpV q, it is ordered by refinement, where for any π 1 , π 2 P DpV, Qq, π 1 ≤ π 2 holds if and only if every block of π 2 is the union of some blocks of π 1 . In [5] we proved the following. Proposition 1.2. Let pV, Qq be a closure system. Then DpV, Qq is a complete lattice with the greatest element ∇ " tV u. If pV, Qq is algebraic and satisfies condition (I 0 ), then DpV, Qq is a complete sublattice of Part(V) if and only if it satisfies condition (I 1 ). Example 1.3. If T " pV, Eq is a finite tree, then the vertex sets of its subtrees form a closure system pV, Qq which satisfies conditions (I 0 ) and (I 1 ). Then DpV, Qq is a finite sublattice of PartpV q, according to Proposition 1.2.
Brought to you by | Tampere University Library Authenticated Download Date | 10/28/16 2:13 PM We prove that DpV, Qq is a Boolean lattice isomorphic to pPpEq, Ďq. Indeed, given S Ď E define πpSq as the equivalence relation on V in which two vertices are equivalent if they are connected in the tree T by a path containing only edges from S. Since the classes of πpSq induce subtrees of T , πpSq is a decomposition in pV, Qq. Then the isomorphism of PpEq to the lattice DpV, Qq is given by the mapping S Þ ÝÑ πpSq.
The following result was proved in full generality in [17] and [5]: Qq is an algebraic closure system satisfying condition pI 1 q, then DpV, Qq is an algebraic semimodular lattice.
Therefore, for an interval system pV, Iq, the lattice DpV, Iq is always an algebraic semimodular sublattice of PartpV q.
Remark 1.5. Let pV, Qq be a closure system satisfying (I 0 ). Then clearly, " ttxu | x P V u is the least element of DpV, Qq, and to any A P Qzt∅u corresponds the decomposition where Ž means the join in the complete lattice DpV, Qq.
A decomposition π " tA i | i P Iu in a closure system pV, Qq is called a strong decomposition if every A i , i P I, is a strong set in pV, Qq. Since the strong decompositions in pV, Qq can be considered also as decompositions in the closure system pV, Sq, they form the complete lattice DpV, Sq whose greatest element is " tV u.
An element a of a lattice L is called standard (see Grätzer [10]), if x^pa _ yq " px^aq _ px^yq holds, for all x, y P L.
The standard elements of L form a distributive sublattice of L denoted by SpLq. An element a is called a neutral element of L, if for any x, y P L the sublattice of L generated by the set ta, x, yu is distributive (see e.g. [10]). Clearly, any neutral element is standard, too. The following result was proved also in [5]: Theorem 1.6. Let pV, Qq be a closure system and S be the family of its strong sets. Then the strong decompositions in pV, Qq are standard elements of DpV, Qq, and DpV, Sq is a distributive sublattice of DpV, Qq and of Part(V).
A set A P Q of a closure system pV, Qq is called fragile if it is the union of two disjoint nonempty members of Q, otherwise A is called nonfragile. This generalizes the concept of fragility studied by Habib and Maurer [11] in the context of module systems of graphs. In view of [5], if pV, Qq is an interval system, then any nonfragile interval A P Q is a strong set.

Completely join-irreducible elements in DpV, Iq
An element p P Lzt0u of a complete lattice L is called completely joinirreducible if for any system of elements x i P L, i P I, the equality p " Ž tx i | i P Iu implies that p " x i for some i P I. Let JpLq stand for the set of completely join-irreducible elements of L. The completely meet-irreducible elements of L are defined dually, consisting a set M pLq. Let us now define a˚:" Ž tx P L | x ă au for any a P Lzt0u, and a˚:" Ź tx P L | x ą au, for any a P Lzt1u. Denoting by ≺ the covering relation in a lattice L, we can observe that p P Lzt0u is completely join-irreducible ô p˚ă p ô p˚≺ p, and m P Lzt1u is completely meet-irreducible ô m ă m˚ô m ≺ m˚.
In this section, we characterize the completely join-irreducible elements in the lattice DpV, Iq of interval decompositions, and we show that they are closely related to the strong sets of pV, Iq. The following result from [5] will be used.
Lemma 2.1. Let π 1 " tB j | j P Ju and π 2 " tA i | i P Iu be two decompositions in a closure system pV, Qq. If π 1 ≺ π 2 holds in DpV, Qq then there exists a unique k P I and K Ď J with at least two elements such that A k " Ť jPK B j and A i P π 1 , for all i P Iztku.
We shall need also the following: Lemma 2.2. Let pV, Qq be a closure system satisfying condition (I 0 ) and π P DpV, Qq. Then π is completely join-irreducible in DpV, Qq if and only if there exists a nonempty closed set A P Q with π " π A and such that A admits a greatest proper decomposition into closed sets.
be an arbitrary decomposition of pA k , Q A k q such that ν ‰ tA k u. Then it is easy to see that ν`" tC t | t P T u Y ttxu | x P V zA k u is a decomposition in DpV, Qq and ν`ă π A k " π. Since π is completely join-irreducible, we get ν`ă π˚. Hence, the partition ν " tC t | t P T u of A is a refinement of the Brought to you by | Tampere University Library Authenticated Download Date | 10/28/16 2:13 PM partition µ " tB j | j P Ju of A. Thus, ν ≤ µ holds in DpA k , Q A k q, and this means that A k admits µ as a greatest proper decomposition.
Conversely, let A P Qzt∅u be a closed set that admits a greatest proper decomposition A " tB j | j P Ju, | J |≥ 2. We prove that π A is a completely join-irreducible element in DpV, Qq.
Then tC t | t P T u is a proper decomposition in pA, Q A q, and hence tC t | t P T u ≤ tB j | j P Ju. Because this result yields µ ≤ tB j | j P Ju Y ttxu | x P V zAu, we deduce pπ A q˚≤ tB j | j P Ju Y ttxu | x P V zAu ă π A , and this implies that π A is completely join-irreducible. Proposition 2.3. Let pV, Iq be an interval system. Then π is a completely join-irreducible element in DpV, Iq if and only if π " π A , where A is an interval admitting a greatest proper decomposition tB j | j P Ju, such that each B j , j P J is a strong set in pV, Iq. If this decomposition of A has at least three blocks, then A is strong.
Proof. Let A P Izt∅u be an interval and µ " tB j | j P Ju, | J |≥ 2, the greatest proper decomposition of A. Then B j P I, for all j P J. If | J |≥ 3, then A cannot be the union of two disjoint nonempty members of I. Therefore, A is nonfragile, and hence it is a strong set according to [5]. In view of Lemma 2.2, to prove our statement it is enough to show that each B j , j P J is strong.
First, assume that | J |≥ 3. Then A is a strong set. Let C P I such that C X B j ‰ ∅, for some j P J. Since A is strong, now C X A ‰ ∅ implies that either A Ď C or C Ĺ A holds. In the first case B j Ď C. If C Ĺ A, then ν " tCu Y ttxu | x P AzCu is a decomposition in pA, I A q and ν ‰ tAu. Hence ν ≤ µ, and this implies C Ď B j , because C X B j ‰ ∅. Therefore, B j is a strong set of pV, Iq.
Let | J |" 2. Then A " B 1 Y B 2 and µ " tB 1 , B 2 u is the greatest proper decomposition in pA, I A q.
Suppose that C X B 1 ‰ ∅, for some C P I. Then also C X A ‰ ∅. If A Ď C then B 1 Ď C holds. Let A Ę C. If C Ď A, then using the same argument as in the above case | J |≥ 3, we obtain C Ď B 1 . If C Ę A then AzC P I t∅u. Because C X A and AzC are nonempty intervals, ρ " tC X A, AzCu is a proper decomposition in pA, I A q. Hence, ρ ≤ µ. Since ρ is a maximal proper partition of the set A, we get ρ " µ. Then B 1 " C X A or B 1 " AzC . The second case is excluded, because C X B 1 ‰ ∅. Thus, we deduce B 1 Ď C.
Brought to you by | Tampere University Library Authenticated Download Date | 10/28/16 2:13 PM Therefore, C X B 1 " ∅ or B 1 Ď C or C Ď B 1 must hold for any C P I, and this means that B 1 is a strong set. The fact that B 2 is strong is proved similarly.
As an immediate consequence of Proposition 2.3 we obtain: Corollary 2.4. Let pV, Iq be an interval system. If π is a completely join-irreducible element in DpV, Iq, then π˚is a strong decomposition.
A lattice L is called geometric, if it is atomistic, semimodular and algebraic. Geometric lattices are also dually atomistic. This follows e.g. from [10, Lemma 391] and semimodularity.
Corollary 2.5. Let pV, Iq be an interval system such that DpV, Iq is of finite length. Then the following assertions are equivalent: (i) DpV, Iq is an atomistic lattice; (ii) DpV, Iq is a geometric lattice; (iii) DpV, Iq is a dually atomistic lattice; (iv) The only strong intervals in I are V , ∅, and the singletons.

Proof. (i)ñ(ii) is clear, because DpV, Iq is an algebraic semimodular lattice, according to Proposition 1.5. The implication (ii)ñ(iii) is obvious, and (iii)ñ(iv) follows from [5, Corollary 3.7].
(iv)ñ(i). Since DpV, Iq is a lattice of finite length, any element of it is a join of some completely join-irreducible elements (see e.g. [1]). Now, assume that pV, Iq has no proper strong intervals, and let π be a completely join-irreducible element of DpV, Iq. Since π ą , we get π˚≥ . Because by Corollary 2.4 π˚‰ tV u is a strong decomposition, in view of Proposition 2.3, we get that any block of π˚is of the form tau, a P V . Then π˚" . Since is the 0-element of DpV, Iq, it follows that π is an atom. Hence DpV, Iq is an atomistic lattice.

Interval systems with distributive decomposition lattices
In this section, we are going to characterize interval systems pV, Iq having a distributive decomposition lattice DpV, Iq. (The question was raised by R. H. Möhring.) First, we introduce some notations and recall some known results.
Let L be a lattice, and for each element a P L define the set Jpaq " tj P JpLq | j ≤ au. Then a " Ž Jpaq, whenever L is of finite length.
Remark 3.1. In [18] it was shown that for all algebraic lattices in which a " Ž Jpaq, for all a P L the following assertions are equivalent: (a) L is distributive. (b) Jpa _ bq " Jpaq _ Jpbq, for all a, b P L.
(c) j ≤ a _ b ô pj ≤ a or j ≤ bq, for any a, b P L and all j P JpLq.

A family of sets
In view of [5], if pV, Iq is an interval system and tA i | 1 ≤ i ≤ nu Ď I is a connected family, then Ť n i"1 A i P I. For any sets A, B let A B " pAzBq Y pBzAq. We say that A and B are overlapping sets if all the relations A X B ‰ ∅, AzB ‰ ∅ and BzA ‰ ∅ hold.
Lemma 3.2. Let pV, Iq be an interval system and A, B P I two overlapping sets. Then the following assertions are equivalent: (ii) There exist some nonempty intervals C, D P I with C Ď AzB, D Ď BzA, such that C Y D P I holds.
Proof. (i)ñ(ii). Take C " AzB and D " BzA. Then (i) yields C Y D P I.
In view of Proposition 2.3, ν " π A , for some A P I having a proper decomposition A " Ť tB j | j P Ju, | J |≥ 2 such that each B j , j P J is a strong set in pV, Iq.
Next, assume that | J |" 2. Then A " B 1 Y B 2 , where B 1 ,B 2 are disjoint and nonempty strong intervals of pV, Iq. Denote by ρ 1 , ρ 2 and ρ A the equivalence relations on V corresponding to the partitions π 1 , π 2 and π A , respectively. Then ρ A ≤ ρ 1 _ ρ 2 implies that for any x P B 1 and y P B 2 there exists a sequence z 0 , z 1 , ..., z n P V , with x " z 0 , y " z n such that, for all 1 ≤ i ≤ n either pz i´1 , z i q P ρ 1 or pz i´1 , z i q P ρ 2 holds. Let us select the elements x P B 1 and y P B 2 so that the length n of the above sequence is as small as possible.
Assume that n " 1. Then px, yq P ρ 1 or px, yq P ρ 2 holds. This means px, yq P C, for some block C P π 1 , or px, yq P D, for some block D P π 2 . Since B 1 ,B 2 are disjoint, C Ď B 1 and C Ď B 2 can not hold simultaneously. If px, yq P C, then C X B 1 ‰ ∅, C X B 2 ‰ ∅ imply B 1 ,B 2 Ď C, because C P I and B 1 ,B 2 are strong intervals. Hence A " B 1 Y B 2 Ď C, and this yields ν " π A ≤ π 1 . Similarly, in the case px, yq P D we obtain A " B 1 Y B 2 Ď D and ν " π A ≤ π 2 .
Finally, we prove that n ≥ 2 is not possible. Let n ≥ 2. Then there are intervals X 1 , X 2 , . . . , X n P π k , k P t1, 2u, such that z i´1 , z i P X i , for all 1 ≤ i ≤ n. Then X i Ď V , 1 ≤ i ≤ n is a connected family of sets, and we get C " X 1 Y¨¨¨Y X n´1 P I and D " X 2 Y¨¨¨Y X n P I. Since the length of the path z 0 , z 1 , . . . , z n is as small as possible, z 1 R B 1 , z n´1 R B 2 , and we infer B 1 XD " ∅ and B 2 X C " ∅.
Indeed, B 1 XD ‰ ∅ would imply B 1 X X l ‰ ∅, for some l P t2, . . . , nu. Then replacing x " z 0 by x 1 P B 1 X X l , we would obtain a sequence x 1 , z l , . . . , z n which connects x 1 P B 1 to y " z n P B 2 and has length less than or equal to n´1, contrary to our assumption. Similarly, we can prove B 2 X C " ∅. Because z 1 P CzB 1 and z n´1 P DzB 2 , we have C Ę B 1 and D Ę B 2 . Since z 0 P C X B 1 and z n P D X B 2 and B 1 , B 2 are strong intervals, we get B 1 Ď C and B 2 Ď D. Hence B 1 Ď CzD, B 2 Ď DzC. Since C X D Ě X 2 ‰ ∅ and CzD, DzC are nonempty sets, C and D are overlapping intervals. As B 1 Y B 2 " A P I, and B 1 Y B 2 Ď pCzDq Y pDzCq " C D by Lemma 3.2 we obtain C D P I, contradicting our assumption.

Further properties of the lattice DpV, Iq
A lattice L of finite length is called a strong lattice if for any join-irreducible element j P JpLq, and for all x P L j ≤ j˚_ x implies j ≤ x.
Obviously, any atomistic lattice L is strong. We say that a lattice L is dually strong, if its dual L pdq is strong. The lattice L is called consistent if, for any j P JpLq and each x P L, the element x _ j is join-irreducible in the interval rx, 1s. If for any j P JpLq and m P M pLq with j ę m j _ m " m˚ô j^m " jh olds true, then L is called a balanced lattice. We say that L satisfies the Kurosh-Ore replacement property for join-decompositions (_-KORP, for short), if for every a P L, and any two irredundant join-decompositions a " j 1 _¨¨¨_ j m and a " k 1 _¨¨¨_ k n , with j 1 ,¨¨¨, j m , k 1 ,¨¨¨, k n P JpLq, each j i can be replaced by a k p so that Remark 4.1. It is well-known that any semimodular lattice of finite length is dually strong (see e.g. Stern [21]). It belongs to the folklore that a lattice L of finite length is balanced if and only if both L and L pdq are strong. Crawley showed [1] that L satisfies _-KORP if and only if L is consistent. Let L be of finite length. As it is noted in [21], from the previous facts together with a result of Walendziak [22,Thm.1] the equivalence of the next assertions follows: (a) L is semimodular and has the _-KORP. (b) L is semimodular and balanced; (c) L is semimodular and consistent; (d) L is semimodular and strong.
Theorem 4.2. Let pV, Iq be an interval system. If the lattice DpV, Iq has finite length, then it is a balanced lattice that satisfies _-KORP.
Proof. Since DpV, Iq is a semimodular lattice of finite length, in order to prove our theorem, in view of Remark 4.1, it suffices only to show that DpV, Iq is strong. Take any j P JpDpV, Iqq and x P DpV, Iq with j ≤ j˚_ x. Because any join-irreducible element of a lattice of finite length is also completely join-irreducible, j˚is a standard element in DpV, Iq, according to Corollary 2.4. Thus we obtain (2) j " j^pj˚_ xq " pj^j˚q _ pj^xq " j˚_ pj^xq.
Since j is join-irreducible and j˚ă j, (2) implies j " j^x. Hence j ≤ x, and this proves that DpV, Iq is strong.
The above theorem has a further consequence for finite interval decomposition lattices.
A tolerance of a lattice L is a reflexive and symmetric relation T Ď L 2 compatible with the operations of L. A block of T is a maximal set B Ď L satisfying B 2 Ď T . Suppose that L is of finite length. Then any block B of T has the form of an interval B " ru, vs, u, v P L, u ≤ v, and the compatibility property of T makes it possible to build a "factor lattice" L{T , whose elements are the blocks of T (see Czédli [2]). T is called a glued tolerance, if it contains all covering pairs of L. Since every intersection of glued tolerances of L is again a glued tolerance of L, there exists a least tolerance ΣpLq comprising all pairs x ≺ y in L, called the skeleton tolerance of L. The lattice L is said to be glued by geometric lattices, if all blocks of ΣpLq are geometric lattices.
Reuter [19] proved (see also [21; Thm. 4.6.8]) that for a finite lattice L, the assertions (a), (b), (c) and (d) of Remark 4.1 are equivalent to the following statement: (e) L is glued by geometric lattices.
Therefore, by Theorem 4.2 we infer: Corollary 4.3. Let pV, Iq be an interval system such that the lattice DpV, Iq of its decompositions is finite. Then DpV, Iq is glued by geometric lattices.