Inductive and divisional posets

We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so-called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. Our first main result is that every divisional poset is factorable. Our second main result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (1984), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third main result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type $A$, $B$ or $C$ with respect to the root lattice is inductive.

A hyperplane arrangement H is a finite set of hyperplanes (1-codimensional affine subspaces) in a finite dimensional vector space V .The intersection poset L(H ) of H is the set of all nonempty intersections of hyperplanes in H , which is often referred to as the combinatorics of H .The arrangement H is called factorable if its characteristic polynomial χ H (t) has all nonnegative integer roots.In this case, we call the roots of χ H (t) the (combinatorial) exponents of H .
An arrangement is called central if every hyperplane in it goes through the origin.A central arrangement H is said to be free if its module D(H ) of logarithmic derivations is a free module (Definition 2.16).A remarkable theorem connecting algebra and combinatorics of arrangements due to Terao asserts that if an arrangement H is free, then it is factorable and its combinatorial exponents coincide with the degrees of the derivations in any basis for D(H ) (Theorem 2.17).Definition 1.1.A property P of arrangements is called a combinatorial property (or combinatorially determined) if for any distinct arrangements H 1 and H 2 in V having the same combinatorics, i.e., their intersection posets are isomorphic L(H 1 ) ≃ L(H 2 ), then H 1 has property P if and only if H 2 has property P .
Based on the factorization theorem mentioned above, Terao conjectured that freeness is a combinatorial property [22,Conjecture 4.138].Terao's conjecture remains open till now even in dimension 3.
A natural approach to the conjecture is to find a significant class of arrangements whose freeness is combinatorially determined.Motivated by the addition-deletion theorem for free arrangements [22,Theorem 4.51], Terao first defined the class of inductively free arrangements in which an arrangement can be built from the empty arrangement by adding a hyperplane one at a time subject to the inductive freeness of both deleted and restricted arrangements, and a divisibility condition on the characteristic polynomials (Definition 2. 19).A notable feature of this class due to Jambu and Terao [16] is that it contains supersolvable arrangements (Definition 2.18), a prominent class of arrangements defined earlier by Stanley [26].Later on, Abe [1] proved a refinement of the addition-deletion theorem, and introduced a proper superclass of inductively free arrangements, the so-called divisionally free arrangements (Definition 2.20).Both inductively and divisionally free arrangements are combinatorially determined, proper subclasses of free arrangements (Remark 2.21).In particular, inductive or divisional freeness is a sufficient condition for the arrangement' factorability.
In recent years, there has been increasing attention towards extending the known properties of hyperplane arrangements to toric arrangements, or more generally, to abelian arrangements.Given an abelian Lie group G = (S 1 ) a × R b (a, b ≥ 0) and a finite set A of integral vectors in Γ = Z ℓ , Liu, Yoshinaga and the third author [19] defined the abelian arrangement A = A (A, G) by means of group homomorphisms from Γ to G (see Section 5 for details).In particular, when G = R (or C) we obtain a real (or complex) hyperplane arrangement, and when G = S 1 (or C × ) this is known as a real (or complex) toric arrangement which describes a finite set of (translated) hypertori in a finite dimensional torus.
We recall some important results of abelian arrangements.In [19], a formula for the Poincaré polynomial of the complement of A when G is noncompact (i.e., b > 0) is given; this generalizes the formulas of Orlik and Solomon [21], and De Concini, Procesi, and Moci [10,20] for complex hyperplane and toric arrangements.(The cohomology ring structure is also known [21,10,8] in the case of hyperplane or toric arrangements.)In [33], the intersection poset (or poset of layers) L(A ) of A is defined as the set of all connected components of intersections of elements in A , and its characteristic polynomial is computed.
It is well-known that the intersection poset of a central hyperplane arrangement is a geometric lattice (Definition 2.2).Bibby and Delucchi [5] recently introduced a more general notion of (locally) geometric posets (Definitions 2.3 and 2.13) and showed that these posets describe the intersection data of abelian arrangements (Theorem 5.2).Furthermore, based on an extension of the concept of lattice modularity, the authors defined the notion of strictly supersolvable posets (Definition 2.9), which is of our particular interest here.It is proved that every strictly supersolvable poset is factorable (Theorem 2.10), which extends the result by Stanley for supersolvable lattices [26].
The first motivation for this work is a pursuit of a theory for "free abelian arrangements".As of this writing, we do not know how to pass from algebraic consideration of freeness of hyperplane arrangements to abelian or just toric arrangements.However, at the purely combinatorial level using only information from the posets, it is possible to define and study the combinatorial structures of abelian arrangements and geometric posets in the same way that inductive freeness and divisional freeness do for hyperplane arrangements and geometric lattices.
In this paper, we give definitions of inductive and divisional posets as subclasses of locally geometric posets (Definitions 3.6 and 3.7).The former is a proper subclass of the latter owing to a deletion-restriction formula for characteristic polynomials (Theorem 3.5 and Proposition 3.8).On the arrangement theoretic side, we define inductive and divisional arrangements in a similar way (Definitions 5.9 and 5.10).We show that an abelian arrangement is inductive (resp., divisional) if and only if its intersection poset is inductive (resp., divisional) (Theorem 5.11).As a consequence, inductiveness and divisionality are combinatorial properties of abelian arrangements (Corollary 5.12).
The second motivation is a contribution to factorability of an abelian arrangement, or more generally, of a locally geometric poset (Definition 2.1).Beyond ranked lattices, there are some reasons for an arbitrary poset to be factorable (e.g., [12]).Our first main result in the paper is that a divisional (in particular, an inductive) poset has this factorability.
Our second main result is a generalization of the classical result of Jambu and Terao [16] mentioned earlier for supersolvable and inductively free arrangements.

Theorem 1.3. If a poset is strictly supersolvable, then it is inductive.
Using the notion of characteristic quasi-polynomial from [17], the third author [32] showed that the toric arrangement defined by an arbitrary ideal of a root system of type A, B or C with respect to the root lattice is factorable.Our third main result is a strengthening of this result.
Theorem 1.4.The toric arrangement defined by an arbitrary ideal of a root system of type A, B or C with respect to the root lattice is inductive.
Finally, we give a discussion on the localization at a layer of an abelian arrangement (Section 6).It is shown that inductive freeness of a hyperplane arrangement is preserved under taking localization [14].We show that it is not the case for an arbitrary abelian arrangement by providing an example of an inductive toric arrangement with a non-inductive localization.Furthermore, this example indicates a rather interesting phenomenon that changing the base group G would turn a non-inductive arrangement into an inductive one -there exists a finite set A of integral vectors whose corresponding hyperplane arrangement A (R) is not inductive but the toric arrangement A (S 1 ) is.
All posets (P, ≤ P ) will be finite and have a unique minimal element 0. All P will also be ranked meaning that for every x ∈ P, all maximal chains among those with x as greatest element have the same length, denoted rk(x).Define the rank of a poset P to be rk(P) := max{rk(x) | x ∈ P}.
The Möbius function µ := µ P of a poset P is the map µ P : P × P −→ Z defined by The characteristic polynomial χ P (t) ∈ Z[t] of P is defined as Definition 2.1.A poset P is factorable if the roots of its characteristic polynomial χ P (t) form a subset of positive integer roots.In this case, we call the roots of χ P (t) the (combinatorial) exponents of P and write exp(P) = {d 1 , . . ., d rk(P) } for the multiset of exponents.Denote by FR the class of factorable posets.
The trivial lattice { 0} is factorable since χ { 0} (t) = 1.In this case, exp({ 0}) = ∅.Let P and Q be posets.A poset morphism σ : P → Q is an order-preserving map, i.e., x ≤ y implies σ(x) ≤ σ(y) for all x, y ∈ P. We call σ a poset isomorphism if σ is bijective and its inverse is a poset morphism.The posets P and Q are said to be isomorphic, written P ≃ Q if there exists a poset isomorphism σ : P → Q.
For a subset T ⊆ P, the join T (resp., meet T ) of T is the set of minimal upper bounds (resp., maximal lower bounds) of elements in T .That is, In particular, when T = {x, y}, we write x ∨ y := T and x ∧ y := T .For x ∈ P, define P ≤x := {y ∈ P | y ≤ x} and P ≥x := {y ∈ P | y ≥ x}.
We call x ∈ P an atom if rk(x) = 1.Denote the set of atoms of P by A(P).For x, y ∈ P, by y covers x, written x <• y, we mean x < y and x ≤ z < y implies x = z.
The poset P is a lattice if |x ∨ y| = 1 and |x ∧ y| = 1 for any x, y ∈ P. In this case by abuse of notation we write, e.g., a = x ∨ y for a ∈ x ∨ y.Definition 2.2.A lattice L is called geometric if for all x, y ∈ L: x <• y if and only if there is an atom a ∈ A(L) with a ≤ x, y = x ∨ a. Definition 2.3.A poset P is called locally geometric if P ≤x is a geometric lattice for every x ∈ P. Remark 2.4.If P is a locally geometric poset, then so are P ≤x and P ≥x for any x ∈ P [5, Remark 2.2.6].Definition 2.5.For any subset B ⊆ A(P), define P(B) to be the poset consisting of the minimal element 0 and all possible joins of the elements in B. We call P(B) the subposet of P generated by B.
Remark 2.6.Note that P(A(P)) = P and every element of P(B) is an element of P. If P is a locally geometric poset (or a lattice), then so is P(B).Definition 2.7.An element x in a geometric lattice L is modular if for all z ≤ x and all y ∈ L: x ∧ (y ∨ z) = (x ∧ y) ∨ z.
Let P be a locally geometric poset.An order ideal in P is a downward-closed subset.The poset P (or an order ideal of P) is called pure if all maximal elements have the same rank.An order ideal Q of P is join-closed if T ⊆ Q implies T ⊆ Q.We denote by max(P) the set of maximal elements in P. ).An M-ideal of a locally geometric poset P is a pure, join-closed, order ideal Q ⊆ P satisfying the following two conditions: (1) |a ∨ y| ≥ 1 for any y ∈ Q and a ∈ A(P) \ A(Q), (2) for every x ∈ max(P), there is some y ∈ max(Q) such that y is a modular element in the geometric lattice P ≤x .An M-ideal Q ⊆ P is called a TM-ideal if condition (1) above is replaced by a stronger condition that such a and y have a unique minimal upper bound, i.e., (1*) |a ∨ y| = 1 for any y ∈ Q and a ∈ A(P) \ A(Q).
Note that the element y in Definition 2.8(2) is necessarily unique since Q is join-closed.The following is a generalization of Stanley's supersolvable lattices [26].Definition 2.9 ([5, Definitions 2.5.1 and 5.1.4]).A locally geometric poset P is supersolvable (resp., strictly supersolvable) if there is a chain, called an M-chain (resp., a TM-chain) In particular, if P is strictly supersolvable with a TM-chain When a poset is geometric, we have the following useful characterization of an M-ideal.
Lemma 2.14 ([5, Theorem 4.1.2]).Let P be a geometric poset, and let Q be a pure, join-closed, proper order ideal of P. Then Q is an M-ideal with rk(Q) = rk(P) − 1 if and only if for any two distinct a 1 , a 2 ∈ A(P) \ A(Q) and every x ∈ a 1 ∨ a 2 there exists a 3 ∈ A(Q) such that x > a 3 .

2.2.
Free arrangements.Now we recall the definition of free arrangements and their related properties.Our standard reference is [22].Throughout this subsection, an "arrangement" means a "central hyperplane arrangement".Let K be a field and let T = K ℓ .Let H be an arrangement in T .Let L(H ) be the intersection poset of H .We agree that T is a unique minimal element in L(H ).Thus L(H ) is a geometric lattice which can be equipped with the rank function rk(X) := codim(X) for X ∈ L(H ) (e.g., [22,Lemma 2.3]).We also define the rank rk(H ) of H as the rank of the maximal element of L(H ).
The characteristic polynomial χ H (t) of H is defined by The empty arrangement ∅ ℓ (or simply ∅) is the arrangement in T consisting of no elements.In particular, ∅ ℓ ∈ FR with exp(∅ ℓ ) = {0 ℓ }.
Let {x 1 , . . ., x ℓ } be a basis for the dual space T * and let A K-linear map θ : S → S is called a derivation if θ(f g) = θ(f )g + f θ(g) for all f, g ∈ S. Let Der(S) be the set of all derivations of S. It is a free S-module with a basis {∂/∂x 1 , . . ., ∂/∂x ℓ } consisting of the usual partial derivatives.We say that a nonzero derivation θ = ℓ i=1 f i ∂/∂x i is homogeneous of degree p if each nonzero coefficient f i is a homogeneous polynomial of degree p [22,Definition 4.2].
The concept of free arrangements was defined by Terao [30,22].
Remark 2.21.Supersolvability, inductive and divisional freeness of central hyperplane arrangements all are combinatorial properties.We give below the relation between the concepts we have defined so far: The first containment is proved by Jambu and Terao [16,Theorem 4.2].The arrangement of a root system of type D ℓ for ℓ ≥ 4 belongs to IF \ SS (e.g., [15,Theorem 6.6]).The second containment follows from the deletion-restriction formula χ H (t) = χ H ′ (t) − χ H ′′ (t) (e.g., [22,Theorem 2.56]).The arrangement defined by the exceptional complex reflection group of type G 31 is known to be divisionally free [1,Theorem 1.6] but not inductively free [13,Theorem 1.1].The third containment is proved by Abe Theorem 5.6] is an example of an arrangement in F \ DF.The fourth containment is Theorem 2.17 by Terao.There are many examples of factorable but not free arrangement, e.g., [11, 3.6].

INDUCTIVE AND DIVISIONAL POSETS
From now on unless otherwise stated, we will assume that P is a locally geometric poset, and set A = A(P) and r = rk(P).Definition 3.1.Fix an atom a ∈ A. Let P ′ := P(A \ {a}) be the subposet of P generated by A \ {a} and define P ′′ := P ≥a .We call (P, P ′ , P ′′ ) the triple of posets with distinguished atom a. Remark 3.2.Note that for each a ∈ A, we have rk(P) = rk(P ′ ) + ǫ(a), where ǫ(a) is either 0 or 1.Indeed, let x ∈ max(P) so that rk(x) = r.If a ≤ x then rk(P ′ ) = r.Otherwise, set We call a ∈ A a separator of P if ǫ(a) = 1.
For each x ∈ P, define Lemma 2.35]).Let P be a geometric lattice.For x, y ∈ P with x ≤ y, let S(x, y) be the set of all subsets B ⊆ A such that A x ⊆ B and max(P(B)) = y.Then Lemma 3.4.Let P be a locally geometric poset.Then the characteristic polynomial χ P (t) strictly alternates in sign, i.e., if Proof.By definition, for each 0 ≤ i ≤ r we have Note that the characteristic polynomial of a geometric lattice strictly alternates in sign (e.g., [27,Corollary 3.5]).Thus (−1) rk(x) µ( 0, x) > 0 since P ≤x is a geometric lattice for every x ∈ P.
We show below that the characteristic polynomials of locally geometric posets satisfy a deletionrestriction recurrence, which is crucial for our subsequent discussion.This formula is already proved for geometric lattices, e.g., see [7,Theorem 1.2.20].The method therein can be readily extended to locally geometric posets, we include here a proof for the sake of completeness.Theorem 3.5.Let P be a locally geometric poset and fix a ∈ A. Then Here ǫ(a) = rk(P) − rk(P ′ ) is either 0 or 1 by Remark 3.2.
Proof.Since P ≤x is a geometric lattice for every x ∈ P, by Lemma 3.3 we have Now we introduce the protagonists of the paper.Definition 3.6.The class IP of inductive posets is the smallest class of locally geometric posets which satisfies (1) { 0} ∈ IP, (2) P ∈ IP if there exists an atom a ∈ A such that P ′′ ∈ IP, P ′ ∈ IP, and χ P ′′ (t) divides χ P ′ (t).
Definition 3.7.The class DP of divisional posets is the smallest class of locally geometric posets which satisfies (1) { 0} ∈ DP, (2) P ∈ DP if there exists an atom a ∈ A such that P ′′ ∈ DP and χ P ′′ (t) divides χ P (t).
Here are the first two important properties of the inductive and divisional posets.Proof.We argue by induction on r = rk(P) ≥ 0. The assertion clearly holds true when r = 0. Suppose r > 0. Since P ∈ IP, there exists an atom a ∈ A such that P ′′ ∈ IP and χ P ′′ (t) divides χ P ′ (t).By the induction hypothesis, P ′′ ∈ DP.Furthermore, by Theorem 3.5, χ P ′′ (t) divides χ P (t).(Note that t ∤ χ P ′′ (t) by Lemma 3.4.)Thus P ∈ DP as desired.Proposition 3.9.Let P, Q be two isomorphic locally geometric posets.Then P ∈ IP (resp., Proof.We show the assertion for IP by double induction on the rank r and number |A| of atoms.The assertion for DP can be proved by induction on the rank r by a similar (and easier) argument.
The assertion is clearly true when r = 0 or |A| = 0. Suppose r ≥ 1 and |A| ≥ 1.Let f : P → Q be a poset isomorphism.Suppose P ∈ IP.Then there exists an atom a ∈ A such that P ′′ ∈ IP, P ′ ∈ IP, and χ P ′′ (t) since the characteristic polynomial is preserved under isomorphism.Remark 3.10.We address here some remarks about the relation of our inductive and divisional posets with some known concepts in literature.
By the induction hypothesis, there exist positive integers d 1 , . . ., d r−1 ∈ Z >0 and an integer d r ∈ Z such that Thus the divisionality of a poset is a sufficient condition for its factorability.The following necessary and sufficient condition for a poset to be divisional is immediate from Definition 3.7.Note that the sum of all exponents of a divisional poset equals the number of atoms.Theorem 3.11.A locally geometric poset P of rank r is divisional if and only if there exists a chain, called a divisional chain Remark 3.12.The converse of Theorem 1.2 is not true in general.Namely, there exists a factorable poset that is not divisional.An example from hyperplane arrangements is already mentioned in Remark 2.21.We give here an example of a poset that is not a lattice.In [12, Example 4.6], the weighted partition poset P := Π w 3 of rank 3 is given with the characteristic polynomial χ P (t) = (t − 3) 2 (see Figure 1).However, P is not divisional because χ P ≥x (t) = t − 2 does not divide χ P (t) for any atom x.The process of constructing an inductive poset P from the trivial lattice (or more generally, from an inductive subposet generated by some atoms) by adding an atom one at a time with the aid of Theorem 3.13 is called an induction table.Each row of the table records the exponents of P ′ and P ′′ and the atom a added at each step.The last row displays the exponents of P.
We will see in Section 7 many examples of posets which are both inductive and geometric arising from abelian arrangements.Figure 2 below depicts an inductive poset that is not geometric.(In particular, it is not the poset of layers of an abelian arrangement by Theorem 5.2.) An inductive poset that is not geometric (left) and an induction table for its inductiveness (right).The elements labelled by x and y do not satisfy the requirement of Definition 2.13.

STRICTLY SUPERSOLVABLE IMPLIES INDUCTIVE
In this section we prove the second main result of the paper (Theorem 1.3).First we need some basic facts of M-ideals.All posets in this section are locally geometric.Proof.First note that A(P) \ A(Q) = ∅ since Q is join-closed.Fix an arbitrary x ∈ max(P).If x ∈ Q, then by Condition 2.8(1) for any a ∈ A(P) \ A(Q) there exists b ∈ a ∨ x such that x < b, a contradiction.We may assume x ∈ P \ Q.Then by Condition 2.8(2), there exists y ∈ max(Q) such that y < x.Thus rk(x) > rk(Q) and hence rk(x) = rk(P).).Let Q be an M-ideal of a poset P with rk(Q) = rk(P)−1.Fix x ∈ P \ Q and let y be an element in max(P) such that x ≤ y.Let y ′ be the unique element in max(Q) such that (y covers y ′ and) y ′ is a modular element in the geometric lattice P ≤y (Definition 2.8).Then x ′ := y ′ ∧ x is the unique element in Q such that x covers x ′ and x ′ is modular in P ≤x .Now we prove a new property of a TM-ideal, extending a well-known property [28, Lemma 1] of a modular element in a finite geometric lattice.
We show that σ is a poset isomorphism whose inverse is exactly τ .First we show that both maps are order-preserving.The assertion for σ is easy.To show the assertion for τ note that for x 1 ≤ R x 2 , if y ∈ max(P) and x 2 ≤ R y, then τ (x 1 ) = y ′ ∧ x 1 and τ (x 2 ) = y ′ ∧ x 2 where y ′ is the unique element in max(Q) such that y ′ is modular in P ≤y .Thus τ (x 1 ) ≤ Q τ (x 2 ) follows easily.Now we show σ x which contradicts the join-closedness of Q.Note that rk(x ∨ a) > rk(x) hence it cannot happen that x > (x ∨ a) ′ .Thus we may assume x ≤ (x ∨ a) ′ .Let y ∈ max(P) so that x ∨ a ≤ y.Let y ′ be the unique element in max(Q) such that y ′ is modular in P ≤y .Then where the second equality follows from the modularity 2.7 of y ′ in P ≤y with x ≤ y ′ , and the third equality follows from Lemma 4.2.
Now we show the assertion for inductiveness by adding the atoms from A(P) \ A(Q) to A(Q) in any order successively with the aid of Theorem 3.13.Write First note that by Lemma 4.1, the poset P is pure.We observe that rk(P i ) = rk(P) = r for every We claim that Q is a TM-ideal of rank r − 1 of P i for every 1 ≤ i ≤ m.(The case i = m is obviously true.)Condition 2.8(1*) is clear.It suffices to show Condition 2.8 (2).First consider i = m − 1. Fix x ∈ max(P m−1 ) ⊆ max(P).Denote L := P ≤x and L m−1 := (P m−1 ) ≤x .Therefore L and L m−1 are geometric lattices sharing top element x.We need to show that there is some y ∈ max(Q) such that y is a modular element in L m−1 .Since Q is a TM-ideal of P, there exists y ′ ∈ max(Q) such that y ′ is a modular element in L. If x > a m then L = L m−1 .We may take y = y ′ .If x > a m then L m−1 = L(A(L) \ {a m }).Since y ′ > a m , we must have that y ′ ∈ L m−1 and y ′ is also a modular element in L m−1 by [16,Lemma 4.6].Again take y = y ′ .Use this argument repeatedly, we may show the claim holds true for every 1 ≤ i ≤ m − 1.Now we show that P i ∈ IP with exp(P i ) = exp(Q) ∪ {i} for every 1 ≤ i ≤ m.Note that by Lemma 4.4, Q ≃ P ≥a for any a ∈ A(P)\A(Q).It is not hard to check that (P 1 , P ′ 1 = Q, P ′′ 1 ≃ Q) is the triple of posets with distinguished atom a 1 , and that a 1 is a separator of P 1 .Hence P 1 ∈ IP with exp(P 1 ) = exp(Q) ∪ {1} by Theorem 3.13.Similarly, (P 2 , P ′ 2 = P 1 , P ′′ 2 ≃ Q) is the triple with distinguished atom a 2 , and that a 2 is not a separator of P 2 .Hence P 2 ∈ IP with exp(P 2 ) = exp(Q) ∪ {2}.Use this argument repeatedly, we may show the claim holds true for every 1 ≤ i ≤ m.The case i = m yields P ∈ IP with exp(P) = exp(Q) ∪ {m} as desired.
Proof of Theorem 1.3.Note that the trivial lattice is inductive.Apply Lemma 4.5 repeatedly to the elements in any TM-chain of a strictly supersolvable poset P.
Example 4.6.The Dowling posets are proved to be strictly supersolvable [5,Example 5.1.8].The poset of layers of the toric arrangement of an arbitrary ideal of a type C root system with respect to the integer lattice is also strictly supersolvable (Theorem 7.9).Hence these posets are inductive by Theorem 1.3.
Remark 4.7.The main result of [16] by Jambu and Terao mentioned in Remark 2.21 is a special case of our Theorem 1.3 when the poset is a geometric lattice.An induction table for a strictly supersolvable poset can easily be constructed using the argument in the proof of Lemma 4.5.
The converse of Theorem 1.3 is not true in general.There are many known examples of central hyperplane arrangements whose intersection lattices are inductive but not (strictly) supersolvable (see e.g., Theorem 7.2).We will see in Corollary 7.15 and Theorem 7.17 new examples from toric arrangements: The poset of layers of the toric arrangement of a type B ℓ root system for ℓ ≥ 3 is inductive, but not supersolvable.That arises from type B 2 depicted in Figure 3 below is inductive and supersolvable, but not strictly supersolvable.
Thus for locally geometric posets, we have proved the following: SSS IP DP FR.
Compared with the relation described in Remark 2.21, supersolvable posets do not form a subclass of inductive posets.The poset of layers of the toric arrangement of a type D 2 root system (the subposet of the poset in Figure 3 generated by The containment IP DP is strict by an example from Remark 2.21.It remains unknown to us whether or not there exists a divisional but not inductive poset among non-lattice, locally geometric posets. exp(P ′ ) a exp(P ′′ ) The toric arrangement of a type B 2 root system with its poset P of layers (left) and an induction table for inductiveness (right).The induction table is derived thanks to Theorem 3.13 which deduces that P is inductive with exponents exp(P) = {2, 2}.In addition, P is supersolvable with the elements of a rank-1 M-ideal colored in blue.However, P is not strictly supersolvable since it has no TM-ideal of rank 1.

INDUCTIVE AND DIVISIONAL ABELIAN ARRANGEMENTS
We first recall preliminary concepts and results of abelian Lie group arrangements, or abelian arrangements for short, following [33,19,4].
We continue to use the notation ∅ ℓ to denote the empty abelian arrangement in When G = R b and Γ = Z ℓ , we obtain A as an arrangement of affine subspaces in T ≃ R bℓ , and in particular a real (or complex) affine hyperplane arrangement when b = 1 (b = 2, resp.).We sometimes call these hyperplane arrangements integral arrangements as the coefficients of the defining equation of any hyperplane are integer.When G = S 1 (or G = C × ≃ S 1 ×R) and Γ = Z ℓ , we obtain an arrangement of real (complex, resp.)translated hypertori or toric arrangement.
For each B ⊆ A , denote We agree that H ∅ := T .The intersection poset L := L(A ) of A is defined by whose elements, called layers, are ordered by reverse inclusion (X ≤ L Y if X ⊇ Y ).Thus L is a pure, ranked poset with a rank function rk(X) = codim(X)/g for every X ∈ L. The minimal element of L is 0 = T , and the atoms of L are the elements of A .
Definition 5.1.Similar to the case of a hyperplane arrangement in an arbitrary vector space, we also refer to the poset L of layers as the combinatorics of the abelian arrangement A .Likewise, a combinatorial property of abelian arrangements is defined analogously to Definition 1.1.
Define rk(A ) to be the rank of L, i.e., the rank of a maximal element in L. The arrangement A is called essential if rk(A ) = ℓ.The characteristic polynomial χ A (t) of A is defined by Here µ := µ L is the Möbius function of L.
Remark 5.3.Note that χ A (t) = t g(ℓ−rk(A )) • χ L (t g ) which has degree gℓ.In particular, if A is essential and g = 1, then χ A (t) = χ L (t).Definition 5.4.Similar to Definition 2.18, we call an abelian arrangement A supersolvable (resp., strictly supersolvable) if its intersection poset L(A ) is supersolvable (resp., strictly supersolvable).Denote also by SS and SSS the classes of supersolvable and strictly supersolvable abelian arrangements, respectively.Definition 5.5.Similar to Definition 2.15, we call an abelian arrangement A factorable if its intersection poset L(A ) is factorable.In this case, we call the roots of χ A (t 1/g ) the (combinatorial) exponents of A and use the notation exp(A ) to denote the multiset of exponents.Denote also by FR the class of factorable abelian arrangements.By Remark 5.3, A ∈ FR if and only if there are positive integers d 1 , . . ., d rk(A ) ∈ Z >0 such that In this case, exp(A ) = {0 ℓ−rk(A ) } ∪ exp(L(A )).
Definition 5.6 ([4, Definitions 13.5 and 13.7]).For each X ∈ L, define The localization A X of A at X is defined as the collection of linear subspaces H α,0 ⊆ Hom(Γ, R g ) with α ∈ A X .For H ∈ A , the restriction A H of A to H is defined by The following is well-known, e.g., used in the proof of [4, Theorem 13.10].
Lemma 5.7.Let A be an abelian arrangement.Let X ∈ L(A ) and Fix H ∈ A , define the deletion A ′ := A \ {H} as an arrangement in T , and A ′′ := A H .We call (A , A ′ , A ′′ ) the triple of arrangements associated to H. From Definition 3.1 and Lemma 5.7, we have that L(A ′ ) = L ′ and L(A ′′ ) = L ′′ .

Theorem 5.8. Let A be a nonempty abelian arrangement and H ∈ A . The following deletionrestriction formula holds
Proof.Apply Theorems 3.5, 5.2 and Remark 5.3.
We are ready to introduce the concepts of inductive and divisional abelian arrangements.
Definition 5.9.The class IA of inductive (abelian) arrangements is the smallest class of abelian arrangements which satisfies (1) for some d ∈ Z.
Definition 5.10.The class DA of divisional (abelian) arrangements is the smallest class of abelian arrangements which satisfies (1) We now show that inductiveness and divisionality depend only on the combinatorics of arrangements.
Proof.We show the assertion for inductiveness by double induction on rk(A ) and |A |.The assertion for divisionality can be proved by induction on rk(A ) by a similar (and easier) argument.
The assertion is clearly true when rk(A ) = 0 or |A | = 0 (i.e., A = ∅).Suppose rk(A ) ≥ 1 and |A | ≥ 1. Suppose A ∈ IA.Then there exists H ∈ A such that A ′′ ∈ IA, A ′ ∈ IA, and Corollary 5.12.The property of being inductive or divisional of an abelian arrangement is a combinatorial property.
Proof.It follows from Proposition 3.9 and Theorem 5.11 above.Remark 5.13.By Remark 4.7 and Theorem 5.11, we have the following: It is an open question to us whether or not the containment IA ⊆ DA is strict.This is related to the question in the last paragraph in Remark 4.7.The example of a hyperplane arrangement that is divisionally free but not inductively free in Remark 2.21 is not an integral arrangement.
An abelian arrangement is inductive if it can be constructed from the empty arrangement by adding an element (= a connected component of a hyperplane) one at a time with the aid of the following "addition" theorem at each addition step.It thus also makes sense to speak of an induction table for an inductive arrangement in a similar way as of inductive posets in Section 3.
Theorem 5.14.Let A = ∅ be an abelian arrangement in T ≃ G ℓ and let Proof.It follows directly from Definition 5.9 and Theorem 5.8.
We complete this section by describing an arrangement theoretic characterization for (strict) supersolvability.
Definition 5.15.Given a subarrangement B of an abelian arrangement A , we say B is an M-ideal of A if L(B) is a proper order ideal of L(A ), and for any two distinct H 1 , H 2 ∈ A \ B and every connected component C of the intersection H 1 ∩ H 2 there exists H 3 ∈ B such that C ⊆ H 3 .More strongly, an M-ideal B is called a TM-ideal of A if (*) for any X ∈ L(B) and H ∈ A \ B the intersection X ∩ H is connected.
Theorem 5.16.Let A be an arrangement of rank r in T ≃ G ℓ .Then A is supersolvable (resp., strictly supersolvable) (Definition 5.4) if and only if there is a chain, called an M-chain (resp., a TM-chain) Proof.Observe that if B ⊆ A , then L(B) is a pure, join-closed ideal of L(A ).Note also that the poset of layers of an abelian arrangement is a geometric poset by Theorem 5.2.Thus by Lemma 2.14, if B is an M-ideal (resp., a TM-ideal) of A , then L(B) is an M-ideal (resp., a TM-ideal) of L(A ) with rk(B) = rk(A ) − 1.Therefore, if there exists an M-chain (resp., a TM-chain) then L(A ) is supersolvable (resp., strictly supersolvable) with an M-chain (resp., a TM-chain) Conversely, if Q is an M-ideal (resp., a TM-ideal) of L(A ) with rk(Q) = rk(A ) −1, then again by Lemma 2.14, the set A(Q) of atoms is an M-ideal (resp., a TM-ideal) of A .Thus if L(A ) is supersolvable (resp., strictly supersolvable), then any M-chain (resp., TM-chain) of L(A ) induces an M-chain (resp., a TM-chain) for A .

LOCALIZATION OF HYPERPLANE AND TORIC ARRANGEMENTS
In this section, we discuss the operation of localizing at a layer of an abelian arrangement in the sense of Definition 5.6.Note from Remark 2.12 that (strict) supersolvability is closed under taking localization: If A ∈ SS (resp.,A ∈ SSS), then A X ∈ SS (resp.,A X ∈ SSS) for every X ∈ L(A ).We will see that in general it is not the case for inductiveness or divisionality.More explicitly, we give an example of an inductive toric arrangement with a non-factorable localization.
First let us recall from the previous section the definition of central (real) hyperplane and toric arrangements as abelian arrangements when the Lie group G is R and S 1 , respectively.Let A be a finite set of integral vectors in Z ℓ .Given a vector α = (a 1 , . . ., a ℓ ) ∈ A, we may define the hyperplane and the hypertorus

and the central toric arrangement
Alternatively, given an integral matrix S ∈ Mat ℓ×m (Z), we may view each column as a vector in Z ℓ so that we may define the central hyperplane and toric arrangements from S as above.
Example 6.1.Let S ∈ Mat 3×6 (Z) be an integral matrix defined as below: Let H S and A S be the central hyperplane and toric arrangements defined by S, respectively.Note that by definition of localization (Definition 5.6) we may write H S = (A S ) X where X denotes the layer (1, 1, 1) ∈ L(A S ).
In fact, H S is linearly isomorphic to the essentialization of the cone of the digraphic Shi arrangement defined by the path 3 → 2 → 1 in [3, Figure 3].The characteristic polynomial of H S is given by χ H S (t) = (t − 1)(t 2 − 5t + 7), which implies that H S is not divisional hence not inductive.However, we may show that A S is inductive with exponents {2, 2, 2}.Let H i denote the (connected) hypertorus defined by the i-th column of the matrix S. The poset of layers of A S and an induction table are given in Figure 4. (Observe also that A S is not locally supersolvable since the localization H S is not supersolvable by the preceding discussion.) The poset of layers of the toric arrangement A S defined by matrix S in (6.1) and an induction table for its inductiveness.
It happens quite often that the hyperplane arrangement defined by a matrix is inductive, but the toric arrangement defined by the same matrix is not (see the next section).Example 6.1 above deduces that the converse is also possible.This is a rare, perhaps counter-intuitive example that toric arrangement could be inductive, while hyperplane arrangement cannot be.

APPLICATION TO TORIC ARRANGEMENTS OF IDEALS OF ROOT SYSTEMS
Our standard reference for root systems is [6].Let Φ be an irreducible (crystallographic) root system in V = R ℓ .Fix a positive system Φ + ⊆ Φ and the associated set of simple roots (base) Define the partial order ≥ on Φ + such that The sequence (t 1 , . . ., t k , . . ., t M ) is called the height distribution of I.The dual partition DP(I) of the height distribution of I is defined as the multiset of nonnegative integers For each Ψ ⊆ Φ + , let S Ψ denote the coefficient matrix of Ψ with respect to the base ∆, i.e., S Ψ = [s ij ] is the ℓ × |Ψ| integral matrix that satisfies Note that the matrix S Ψ depends only upon Φ. Definition 7.1.Following the previous section, we define A Ψ := A S Ψ (Φ) and H Ψ := H S Ψ (Φ) as the central toric and hyperplane arrangements defined by S Ψ respectively.We call these arrangements the arrangements with respect to the root lattice.25,2,15,24,9]).If I is an ideal of an irreducible root system Φ, then H I is inductive with exponents DP(I).Moreover, In contrast to the hyperplane arrangement case, the toric arrangement A I is not factorable for most cases even when I = Φ + .It is known that the characteristic polynomial of the central toric arrangement defined by an arbitrary matrix S coincides with the last constituent of the characteristic quasi-polynomial χ quasi S (q) defined by S [19, Corollary 5.6].Furthermore, an explicit computation shows that the last constituent of χ quasi S Φ + (q) factors with all integer roots if and only if Φ is A ℓ , B ℓ or C ℓ [18,29].Thus, A Φ + is factorable if and only if Φ is of one of these three types.
Even more is true: If I is an ideal of an irreducible root system of type A, B or C, then A I is factorable whose combinatorial exponents can be described by the signed graph associated to I [32].Our third main result Theorem 1.4 strengthens this result.Furthermore, we give an explicit description of the exponents of A I derived from an explicit induction table.This description turns out to be equivalent to the ones in [32].We also give a characterization for supersolvability of A Φ + when Φ is of type B (Theorem 7.17 The proof for the type A case in Theorem 1.4 is a simple consequence of Theorem 7.2, which we give below. Corollary 7.3.If I is an ideal of a root system of type A, then the toric arrangement A I with respect to the root lattice is strictly supersolvable (equivalently, supersolvable) hence inductive with exponents DP(I).

Proof. It is not hard to see that for any
which is a geometric lattice.By Remark 2.12, its supersolvability and strict supersolvability are equivalent.Moreover, A I is indeed supersolvable with exponents DP(I) by Theorem 7.2.
Hence we are left with the computation on types B and C. First we need a construction of root systems of these types via a choice of basis for V following [6,Chapter VI,§4].
Let E := {ǫ 1 , . . ., ǫ ℓ } be an orthonormal basis for V .For ℓ ≥ 1, is an irreducible root system of type B ℓ .We may choose a positive system Define α i := ǫ i − ǫ i+1 for 1 ≤ i ≤ ℓ − 1, and α ℓ := ǫ ℓ .Then ∆(B ℓ ) = {α 1 , . . ., α ℓ } is the base associated to Φ + (B ℓ ).We may express For Ψ ⊆ Φ + (B ℓ ), write T Ψ = [t ij ] for the coefficient matrix of Ψ with respect to the basis E. The matrices T Ψ and S Ψ are related by T Ψ = P (B ℓ ) • S Ψ , where P (B ℓ ) is an unimodular matrix of size ℓ × ℓ given by Similarly, an irreducible root system of type C ℓ for ℓ ≥ 1 is given by The coefficient matrices of Φ + w.r.t.∆ and E are given by The coefficient matrix of Φ + w.r.t.∆ is S Φ + above with rows switched (this is not the case when ℓ ≥ 3).The coefficient matrix of Φ + w.r.t.E = {ǫ 1 , ǫ 2 } is given by Definition 7.5.Let Φ = B ℓ or C ℓ .For Ψ ⊆ Φ + , denote by A T Ψ and H T Ψ the central toric and hyperplane arrangements defined by the matrix T Ψ , respectively.We call these arrangements the arrangements with respect to the integer lattice.
Remark 7.6.Since the matrix P (B ℓ ) is unimodular, for every Ψ ⊆ Φ + (B ℓ ) we have an isomorphism of posets of layers: L(A Ψ ) ≃ L(A T Ψ ) (see e.g., [23, §5]).However, det P (C ℓ ) = 2.In general, A positive system Φ + (A ℓ−1 ) of an irreducible root system Φ of type A ℓ−1 for ℓ ≥ 2 can be defined as the ideal of To describe the exponents of A I when Φ is B ℓ or C ℓ , we need information from the signed graph associated to I. Definition 7.7.Let Φ = B ℓ or C ℓ .For Ψ ⊆ Φ + and 1 ≤ i ≤ ℓ, define the subset For α ∈ E i , let H α denote the hypertorus defined by α.For example, α = ǫ i + ǫ j defines the hypertorus H α = {t i t j = 1}.We then define the subarrangement In the language of signed graphs (e.g., following [34, §5]), the elements in E + i (Ψ) and E − i (Ψ) correspond to the positive and negative edges of the signed graph defined by Ψ, respectively.
It is not hard to see that for each ideal I of Φ + (B ℓ ) or Φ + (C ℓ ), the elements of the dual partition DP(I) can be expressed in terms of b i (I)'s and vice versa.However, the numbers b i 's are a bit more convenient for our subsequent discussion.7.1.Type C. We first present the results on type C as the proofs are simpler than those on type B. We begin by proving a lemma which serves as a template for some arguments later.It remains to show that for any two distinct H 1 , H 2 ∈ A \ D and every connected component C of the intersection H 1 ∩ H 2 , there exists H 3 ∈ D such that C ⊆ H 3 .We consider three main cases, the remaining cases are similar to one of these.(a) Assume H 1 = {t 1 t j = 1} (i.e., ǫ 1 + ǫ j ∈ I) and Then by the definition of an ideal we must have ǫ j + ǫ k ∈ D (since ǫ 1 + ǫ j > ǫ j + ǫ k ).Hence (c) Assume H 1 = {t 1 = 1} (i.e., 2ǫ 1 ∈ I) and H 2 = {t 1 t j = 1} for j > 1.Then H 3 := {t j = 1} ∈ D (since 2ǫ 1 > 2ǫ j ).Moreover, H 1 ∩ H 2 is connected and This concludes that D is a TM-ideal of A as desired.
Theorem 7.9.Let I ⊆ Φ + (C ℓ ) be an ideal.Define Then the toric arrangement A T I with respect to the integer lattice is strictly supersolvable with exponents exp(A In particular, A s can be identified with By Theorem 5.16, it suffices to show that the chain is a TM-chain of A .A similar argument as in the proof of Lemma 7.8 shows that A i+1 is a TM-ideal of A i for each n ≤ i ≤ ℓ − 1.
Thus A ∈ SSS with the desired exponents.
Recall the definitions of the parameters n ≤ s in Theorem 7.9.
Theorem 7.10.Let I ⊆ Φ + (C ℓ ) be an ideal.Then the toric arrangement A I with respect to the root lattice is inductive with exponents Case 1.First we prove the assertion when s = 1.In this case, I = Φ + .We show that A ∈ IA with the desired exponents by induction on ℓ.The case ℓ = 1 is clear.
Suppose ℓ ≥ 2. Let δ := 2ǫ 1 = 2 1≤k<ℓ α k + α ℓ denote the highest root of Φ + .Define . By the induction hypothesis, D ∈ IA with exponents Denote A ′ := A \ {H δ }.Note that A ′ \ D consists of the hypertori defined by the roots in E 1 (Φ + ).These roots are given by Using a similar argument as in the proof of Lemma 7.8, we may show that D is an M-ideal of A ′ .Moreover, it is indeed a TM-ideal since Condition 5.15(*) is satisfied because the coefficient at the simple α 1 of all roots in E 1 (Φ + ) is 1, while that of the roots in D is 0. Apply Lemma 4.5 for L(D) and L(A ′ ) we have that A ′ ∈ IA with exponents ) Thus by Theorem 7.9, A H δ ∈ IA with exponents exp(A H δ ) = {2(ℓ − i)} ℓ−1 i=1 .Apply Theorem 5.14, we know that A ∈ IA with the desired exponents Case 2. Now we prove the assertion when s > 1.The set can be identified with Φ + (C ℓ−s+1 ).By Case 1 above, P := A J ∈ IA with exponents exp(P) = {2(ℓ − i)} ℓ−1 i=s ∪ {ℓ − s + 1}.Using a similar argument as in Case 1, we may show that the sets E i (I) for n ≤ i ≤ s − 1 give rise to a chain of TM-ideals for A starting from P. Applying Lemma 4.5 repeatedly, we may conclude that A ∈ IA with the desired exponents.

Type B.
The restriction of an ideal toric arrangement of type B is in general not an ideal toric arrangement.We need an extension of the ideals so that the corresponding arrangements contain sufficient deletions and restrictions in order to apply the addition theorem 5.14 to guarantee the inductiveness.
Proof.Denote A := A T I(p) .We may write We show that A ∈ IA with the desired exponents by induction on ℓ.If ℓ ≤ 2, then A is always strictly supersolvable except when p = 3 and I = I(3) = Φ + (B 2 ).In which case, A is indeed inductive with exponents {2, 2} by Figure 3. Now suppose ℓ ≥ 3. Since ǫ 1 + ǫ m ∈ I, we must have ǫ 2 + ǫ m ∈ I. Define Then J can be regarded as an ideal of Φ + (B ℓ−1 ) (via x i → x i−1 ) with m(J ) ≤ m(I) − 1.Also, .Now we show that adding the p − m hypertori t 1 t p−1 = 1, t 1 t p = 1, . . ., t 1 t m = 1 to D in any order and applying Theorem 5.14 to each addition step, we are able to conclude that A ∈ IA with the desired exponents.Since 2ℓ − m = b 1 , it suffices to show that the restriction at each addition step is inductive with exponents {2ℓ − p + 1} ∪ {b i (I)} ℓ−1 i=2 .Indeed, the restriction at each step has the form P ∪ {H k } where H k denotes the hypertorus t k = −1 for some m ≤ k ≤ p − 1. Fix m ≤ k ≤ p − 1.Note that ǫ i + ǫ k ∈ I ⊆ J (p − 1) for all 1 < i = k since ǫ 1 + ǫ k ∈ I. Thus, the restriction (P ∪ {H k }) H k can be identified with the arrangement A T R (1) , where R(1) is the extension with parameter p = 1 of an ideal R of Φ + (B ℓ−2 ) (via x i → x i−1 (2 ≤ i < k) and x i → x i−2 (k < i ≤ ℓ)) with b ± i (R) = b ± i+1 (I) − 1 for 1 ≤ i ≤ ℓ − 2. (Note that the equations b ± i (R) = b ± i+1 (I) − 1 for k − 1 ≤ i ≤ ℓ − 2 follow from the fact that ℓ−2 i=k−1 (E i (R) ∪ {2ǫ i }) is a root system of type C.) Now using a similar argument as in the proof of Theorem 7.9, we know that (P ∪ {H k }) H k is strictly supersolvable hence inductive with exponents can be identified with the extension J (p − t + 1), where J is an ideal of Φ + (B ℓ−t+1 ) with m(i) < p − t + 1 for all 1 ≤ i ≤ ℓ − t + 1.By Lemma 7.12, P := A T J (p−t+1) ∈ IA with exponents exp(P) = {2ℓ − p − t + 2} ∪ {b i (I)} ℓ−1 i=t .Using a similar argument as in the proof of Lemma 7.8, we may show that the sets E i (I) for n ≤ i ≤ a − 1 and E i (I) ∪ {ǫ i } for a ≤ i ≤ t − 1 give rise a chain of TM-ideals for A starting from P. (Note that by definition m(i) ≥ p for all s ≤ i ≤ t − 1.) Applying Lemma 4.5 repeatedly we may conclude that A ∈ IA with the desired exponents.Indeed, the sets above contribute to exp(A ) the exponents b i for n ≤ i ≤ a − 1 and b i + 1 for a ≤ i ≤ t − 1.
Recall from Remark 7.6 that A Ψ and A T Ψ have isomorphic poset of layers for every Ψ ⊆ Φ + (B ℓ ).Proof.If ǫ i / ∈ I for all 1 ≤ i ≤ ℓ, then I can be regarded as an ideal of Φ + (A ℓ−1 ).Thus A I is indeed strictly supersolvable hence inductive by Corollary 7.3.Otherwise, we know that A T I is inductive which follows from Theorem 7.13 by letting p = ℓ + 1.
In contrast to the inductiveness, the toric arrangement of a root system of type B ℓ is not supersolvable for most cases.
If ℓ ≥ 4, then L ≤x is not supersolvable by Remark 2.21.Therefore, L is not locally supersolvable hence not supersolvable.
When ℓ ≤ 3, however, L ≤x is always supersolvable.We need a direct examination for the supersolvability of L. The assertion is clear when ℓ = 1.The case ℓ = 2 is shown in Figure 3. Now we show that L is not supersolvable (though locally supersolvable) when ℓ = 3 by showing that L does not have an M-ideal of rank 2.
Suppose to the contrary that such an M-ideal exists and call it Q.Denote H + ij := {t i t j = 1} and H − ij := {t i t −1 j = 1}.First, notice that a rank-2 element of the form t i = t j = −1 covers exactly two atoms, namely H + ij and H − ij .If these atoms are not in Q, then Lemma 2.14 fails.Hence, at least one of them belongs to Q for every pair of indices i = j ∈ {1, 2, 3}.Moreover, we may deduce that exactly one of H + ij and H − ij belongs to Q. Otherwise, the join H + ij ∨ H − ij ∨ H where H is either H + jk or H − jk for k / ∈ {i, j} contains an element of rank 3, which contradicts the join-closedness of Q.
We consider two main cases, the remaining cases are similar to one of these.

Lemma 4 . 1 .
If a poset P has an M-ideal Q with rk(Q) = rk(P) − 1, then P is necessarily pure.

Lemma 4 . 2 (
[5, Lemma 2.4.6]).Let Q be an M-ideal of a poset P with rk(Q) = rk(P) − 1 and let a ∈ P. Then a ∈ A(P) \ A(Q) if and only if y ∧ a = 0 for all y ∈ max(Q).

Lemma 4 . 4 .
If Q is a TM-ideal of a poset P with rk(Q) = rk(P)−1, then for any a ∈ A(P)\A(Q) there is a poset isomorphism Q ≃ P ≥a .Proof.Fix a ∈ A(P) \ A(Q) and denote R := P ≥a .Owing to Definition 2.8(1*) and Proposition 4.3, two poset maps σ and τ below are well-defined:

Lemma 7 . 8 .
Let I ⊆ Φ + (C ℓ ) be an ideal such that E 1 (I) = ∅.DefineD := I \ (E 1 (I) ∪ {2ǫ 1 }) if 2ǫ 1 ∈ I, I \ E 1 (I) otherwise.Then D can be regarded as an ideal of Φ + (C ℓ−1 ) andA T D is a TM-ideal of A T I .Proof.The first assertion is clear via the transformation x i → x i−1 for 2 ≤ i ≤ ℓ.Denote A := A T I and D := A T D .There do not exist X ∈ L(D) and Y ∈ L(A ) \ L(D) such that X ⊆ Y since the defining equations of any X ∈ L(D) do not involve t 1 .Therefore, L(D) is a proper order ideal of L(A ).Note also that the power of variable t 1 in the defining equation of any H ∈ A \ D is equal to 1.This shows Condition 5.15(*).
Definition 7.7 for the definition of b i 's.) Proof.Denote A := A T I .Note that n ≤ s and b i = 0 for 1 ≤ i < n.If 2ǫ i / ∈ I for all 1 ≤ i ≤ ℓ, then I can be regarded as an ideal of Φ + (A ℓ−1 ) by Remark 7.6.Thus, L(A I ) ≃ L(A T I ).By Corollary 7.3, A ∈ SSS with exponents DP(I) = {b 1 , . . ., b ℓ }.Now we may assume 1 ≤ n ≤ s ≤ ℓ.Then 2ǫ i ∈ I and E i (I) = ∅ for all s ≤ i ≤ ℓ.Define
An arrangement H is called factorable if its intersection poset L(H ) is factorable (Definition 2.1).In this case, we also call the roots of χ H (t) the (combinatorial) exponents of H and use the notation exp(H ) to denote the multiset of exponents.Denote also by FR the class of factorable arrangements.Notation.If an element e appears d ≥ 0 times in a multiset M, we write e d ∈ M.If H ∈ FR, then exp [22,d on this, Terao conjectured that freeness is a combinatorial property[22, Conjecture  4.138].Although Terao's conjecture is still open, there are some subclasses of free arrangements that are known to be combinatorially determined.Definition 2.18.An arrangement H is called supersolvable if its intersection lattice L(H ) is supersolvable (Definition 2.9).Denote also by SS the class of supersolvable (= strictly supersolvable) central hyperplane arrangements.Fix H ∈ H , define the deletion H ′ := H \ {H} and restriction H ′′ IP.(2) A central hyperplane arrangementH in V = K ℓ isinductively free (resp., divisionally free) in Definition 2.19 (resp., 2.20) if and only if the (geometric) intersection lattice L(H ) of H is inductive (resp., divisional).In particular, IP DP which follows from Remark 2.21.Now we give a proof of the first main result of the paper.Proof of Theorem 1.2.We need to show that if P ∈ DP with r = rk(P) ≥ 1, then there are positive integers d 1 , . . ., d r ∈ Z >0 such that Definition 1.2.21] defined the class IL of inductive lattices to be the smallest class of geometric lattices which satisfies: (1) { 0} ∈ IL and (2) P ∈ IL if there exists an atom a ∈ A such that P ′′ ∈ IL, P ′ ∈ IL, and χ P ′′ (t) divides χ P ′ (t).Thus for a geometric lattice P, we have that P ∈ IL if and only if P ∈