WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS TO VERTEX COVER IDEALS

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1 Mohammadi, F. and Moradi, S. Osaka J. Math. 47 (200), WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS TO VERTEX COVER IDEALS FATEMEH MOHAMMADI and SOMAYEH MORADI (Received February 3, 2009) Abstract In this paper we extend the concept of weakly polymatroidal ideals to monomial ideals which are not necessarily generated in one degree, and show that any ideal in this class has linear quotients. As an application we study some vertex cover ideals of weighted hypergraphs. Introduction Hibi and Kokubo in [2] introduced weakly polymatroidal ideals as a generalization of polymatroidal ideals. They considered ideals which are generated in the same degree. In this paper we extend their definition to ideals which are not necessarily generated in one degree. We show that these ideals have linear quotients, and consequently are componentwise linear. Let R K [x,, x n ] be the polynomial ring in n variables over a field K and I R be a monomial ideal. Recall that I has linear quotients, if there exists a system of minimal generators f, f 2,, f m of I such that the colon ideal ( f,, f i )Ï f i is generated by a subset of x,, x n for all i. Ideals with linear quotients were introduced by Herzog and Takayama in [9]. For a homogeneous ideal I R and d, we denote by (I d ) the ideal generated by all forms of degree d in I. The ideal I is called componentwise linear if for each d, (I d ) has a linear resolution. Componentwise linear ideals were introduced by Herzog and Hibi in [0] and they proved that the Stanley Reisner ideal I ½ is componentwise linear if and only if I ½ is sequentially Cohen Macaulay (see [0], [8]). In general it is hard to prove that an ideal is componentwise linear. A criteria for an ideal being componentwise linear are given in [5]. On the other hand, if an ideal has linear quotients, then it is componentwise linear, as shown in [22]. If, in addition, I is a monomial ideal, then I even has componentwise linear quotients, see [23]. Due to these facts one would expect that a weakly polymatroidal ideal is componentwise weakly polymatroidal. However, Example.8 shows that this is in general not the case. Nevertheless, if all generators of the weakly polymatroidal ideal are of same degree, then all components of the ideal are weakly polymatroidal. This is a consequence of The Mathematics Subject Classification. Primary 3D02, 3P0; Secondary 6E05.

2 628 F. MOHAMMADI AND S. MORADI orem.6 where we show more generally that if I is a weakly polymatroidal ideal and m is the graded maximal ideal of polynomial ring, then mi is again weakly polymatroidal. In Section 2 we study classes of ideals which are weakly polymatroidal. The ideals we study are vertex cover ideals of weighted hypergraphs, introduced in [6]. Let V be a finite set and E be a finite collection of nonempty subsets of V. Then H (V, E) is called a hypergraph and the elements of E are called the edges of H. A vertex cover of H is a subset of V which meets every edge of H. Any vertex cover of V can be considered as a (0, ) vector c (c,, c n ), where A¾E(H) c i for all A ¾ E(H). Given a hypergraph H and an integer valued function Ï E(H) Æ, A A, the pair (H, ) is called a weighted hypergraph. For k ¾ Æ, a k-cover is defined as a vector c¾æ n that satisfies the condition i¾a c i k A for any A¾ E(H). For a nonempty subset A a,, a r of V let P A (x a,, x ar ). For a weighted hypergraph (H, ), the ideal A k (H, ) A¾E(H) Pk A A is called the ideal of k-covers of (H, ). The graded vertex cover algebra A(H, ) is defined as k 0 A k(h, ). If A for any A ¾ E(H), then A(H, ) is denoted by A(H). The fundamental question we want to address in this paper is the following: for which weighted hypergraph (H, ) are all its ideals of k-covers componentwise linear? We call a hypergraph with this property uniformly linear. To classify all uniformly linear hypergraphs is quite hopeless. It is known by a result of Francisco and Van Tuyl that the ideal -covers of any chordal graph is componentwise linear, see [8]. However it seems to be unknown whether chordal graphs are uniformly linear. In this paper we consider the following classes of uniformly linear weighted hypergraphs. Actually, we show in all these cases that for any k the ideal of k-covers is either weakly polymatroidal or componentwise weakly polymatroidal. () G is a Cohen Macaulay bipartite graph (see Theorem 2.2). (2) Let H be a hypergraph on the vertex set [n] satisfying one of the following conditions: (a) H has only two facets (see Theorem 2.3). (b) E(H) K, J,, J s and there exists an integer t such that s i J i J i J j [t] for all i j and K J i for i,, s (see Theorem 2.4). (c) E(H) K, J,, J s with J i J j [n] for all i j (see Theorem 2.5). Herzog and Hibi in [9] showed that the ideal of k-covers of a Cohen Macaulay bipartite graph has linear quotients. The result in () is inspired by their work. Francisco and Van Tuyl in [7] among other results proved that the ideals of k-covers of hypergraphs in (a), and (c) in the case that K are componentwise linear.. Weakly polymatroidal ideals Let R K [x,, x n ] be a polynomial ring over the field K. For any monomial ideal I, let G(I ) be the minimal set of generators of I and m(u) be the greatest integer i for which x i divides u. For u x a x a n n, we denote a i by deg xi (u). For a and b in N n we have a lex b if and only if the left-most nonzero entry in a b is positive.

3 WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS 629 DEFINITION.. A monomial ideal I is called weakly polymatroidal if for every two monomials u x a x a n n lex Ú x b x b n n b t and a t b t, there exists j t such that x t (Ú x j ) ¾ I. in G(I ) such that a b,, a t The monomial ideal I is called stable if, for any monomial u in I and i m(u) one has x i (u x m(u) ) ¾ I. From the above definition one can see that each stable ideal is weakly polymatroidal. In [] weakly stable ideals are defined as a generalization of stable ideals and it was shown that every weakly stable ideal generated in degree d has a linear resolution. Also, in that paper an explicit formula for the Betti numbers of such ideals were given. A squarefree monomial ideal I is called weakly stable if for every squarefree monomials u ¾ I and u ¼ u m(u) the following condition ( ) holds: ( ) For every integer l Supp(u) such that l m(u ¼ ), there exists an integer i ¾ Supp(u) with i l such that x l (u x i ) ¾ G(I ). From the definition of weakly stable ideals, it is easy to see that each weakly stable ideal which is generated in one degree is weakly polymatroidal. The following example shows that the converse of the above statement is not true. EXAMPLE.2. The ideal I (x x 3 x 5, x x 3 x 6, x x 4 x 6, x 2 x 4 x 6 ) is weakly polymatroidal, but there exists no permutation of variables such that I is weakly stable. First we prove the following theorem for weakly polymatroidal ideals. Theorem.3. Any weakly polymatroidal ideal I has linear quotients. Proof. order with respect to x x 2 x n. We show that I has linear quotients with respect to u,, u m. Let u i and u j be in G(I ) and u i lex u j. We can assume that u i Let G(I ) u,, u m, where u u 2 u m in the lexicographical x a x a n n and u j x b x b n n and for some t, t n we have a b,, a t b t and a t b t. Therefore there exists l t such that x t (u j x l ) ¾ I. Thus the set A u k Ï x t (u j x l ) ¾ (u k ) is nonempty. Let u s ¾ A be the unique element such that for any u k ¾ A (k s), we have either deg(u k ) deg(u s ) or deg(u k ) deg(u s ) and u k lex u s. Assume that x t (u j x l ) u s h for some h ¾ R. If x t h, then u j u s h ¼ for some h ¾ ¼ R, which is a contradiction by the assumption that u j ¾ G(I ). So we have x b t t u s. We claim that u s lex u j. By contradiction assume that u s lex u j. Let u s u s, one x c x c n n, c b,, c r b r and c r b r for some r n. Since x b t t has r t. Then from the definition of weakly polymatroidal, one has Û u s x r x k ¾ I for some k r. Since r l, x r h and so x k h x r ¾ R. From Û(x k h x r ) x t (u j x l ), Û lex u s and deg(û) deg(u s ), we have Û G(I ). Let Û u s ¼h ¼ for some s ¼ m

4 630 F. MOHAMMADI AND S. MORADI and h ¾ ¼ R. Then deg(u s ¼) deg(û) deg(u s ) which is a contradiction, since u s ¼ ¾ A. Therefore one has u s h ¾ (u,, u j ). Since (u s h Ï u j ) x t, the proof is complete. From [23, Theorem 2.7] we have the following Any weakly polymatroidal ideal has componentwise linear quo- Corollary.4. tients. REMARK.5. Let I u, u 2,, u r be a weakly polymatroidal ideal with respect to the ordering u lex u 2 lex lex u r on G(I ). With the notation in the proof of above theorem for u i lex u j there exists l t, s j and h ¾ R such that x t (u j x l ) u s h. Therefore, for each k r, the ideal I ¼ u,, u k is again weakly polymatroidal. It is known that the product of any two polymatroidal ideals is again polymatroidal, see [4, Theorem 5.3]. This is not the case for weakly polymatroidal ideals. The ideal I (x 2 x 2, x 2 x 3, x x 2 3, x 2x 2 3, x x 3 x 4 ) is weakly polymatroidal, but I 2 does not even have linear resolution. However, we have Theorem.6. Let I be a weakly polymatroidal ideal and m be the maximal ideal of R. Then mi is again weakly polymatroidal. Proof. order with respect to x x 2 x n. Let Û and Û 2 be elements of G(mI ) and Û lex Û 2. Consider the sets A u ¾ G(I )Ï Û x i u for some i n and Let G(I ) u,, u m, where u u 2 u m in the lexicographical B Ú ¾ G(I )Ï Û 2 x j Ú for some j n. Let u ¾ A and Ú ¾ B be the greatest elements in A and B with respect to lex order, respectively. Assume that Û x i u x a x a n n and Û 2 x j Ú x b x b n n, a i b i for i t and a t b t. One has u x a x a i i x a n n and Ú x b x b j j x b n n. First we consider the case t i. We have u lex Ú. If t j, then j is the smallest index with deg x j (u) deg x j (Ú). Then x j (Ú x l ) ¾ I for some l j and we have x j Ú x l Û for some Ú lex Û ¾ I. Since x l Û is in G(mI ), one has Û ¾ G(I ) which is a contradiction by the way of choosing Ú. So t j and t is the smallest index with deg xt (u) deg xt (Ú). Therefore x t (Ú x l ) ¾ I for some l t and so x t (x j Ú x l ) x j (x t Ú x l ) is in mi. Let t i. If j i, then the result is clear. If j i, then u lex Ú, so j is the first index such that a j deg x j (u) deg x j (Ú) b j. Therefore x j (Ú x l ) is in I for some l j and we have x j Ú x l Û for some Û ¾ G(I ), Û lex Ú, which is a contradiction. Therefore, one can assume that j i. If i t, then Ú lex u, since b i deg xi (Ú) for deg xi (u) a i. Then there exists l i such that x i (u x l ) ¾ I. So ux i Ûx l some Û ¾ G(I ), where Û lex u which is a contradiction. Let t i. Then x t Ú x t (x j Ú x j ) ¾ I and j t, which completes the proof.

5 WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS 63 As an immediate consequence of the above theorem we have Corollary.7. Let I be a weakly polymatroidal ideal generated by monomials in one degree. Then I is componentwise weakly polymatroidal. Proof. Assume that the minimal generators of I are of degree d. Then for any j 0 we have (I d j ) m j I, where m is the maximal ideal of R. Therefore by the above theorem (I d j ) is weakly polymatroidal. A weakly polymatroidal ideal for which the minimal generators are not of the same degree, is not necessarily componentwise weakly polymatroidal. The following example shows this fact. EXAMPLE.8. The ideal I (x x 3, x 2 x 3, x x 4 x 5, x 2 x 4 x 5 ) (x, x 2 ) (x 3, x 4 ) (x 3, x 5 ) is weakly polymatroidal. But there exists no permutation of variables such that (I 3 ) is weakly polymatroidal. 2. Some applications As the first application we consider the ideal of k-covers of a Cohen Macaulay bipartite graph G. Let P be a finite poset associated to G, see [, Theorem 3.4], and let J(P) be the distributive lattice consisting of all poset ideals of P, ordered by inclusion. Recall that a subset I P is a poset ideal of P if for all x ¾ I and y ¾ P such that y x, one has y ¾ I. Let P p,, p n and S K [x,, x n, y,, y n ] be the polynomial ring in 2n variables over a field K. To each poset ideal I of P we associate the monomial u I p i ¾I x i p i ¾PÒI y i. The squarefree monomial ideal of S generated by all monomials u I with I ¾ J(P) is denoted by H P. The ideal H P is in fact the ideal of -covers of G. Herzog and Hibi in [, Theorem 3.4] proved that a bipartite graph G with vertex partition X Y is Cohen Macaulay if and only if there exists a labeling on the vertices X x,, x n and Y y,, y n such that: (i) x i y i are edges for i,, n; (ii) if x i y j is an edge, then i j; (iii) if x i y j and x j y k are edges, then x i y k is an edge. In [9] it was shown that the powers of H P have linear quotients. In the following we show that the powers of H P are even weakly polymatroidal. The following lemma was proved in [20, p. 99]. For the convenience of the reader we give a proof of it. Lemma 2.. Let G be a Cohen Macaulay bipartite graph as above. Then each element of the set of minimal generators of the ideal of k-covers of G, A¾E(G) Pk A, can be written as u B u Bk, where B i ¾ J(P) and B k B.

6 632 F. MOHAMMADI AND S. MORADI Proof. By [6, Theorem 5.] we know that the vertex cover algebra of G is standard graded, therefore it follows that A¾E(G) Pk A ( A¾E(G) P A) k (H P ) k. Let u A,, u Ak ¾ H P. From [4, Theorem 2.2] we have u Ai A j and u Ai A j are elements of H P for any i and j. Then we can write u Ai u A j u Ai A j u Ai A j. By induction we show that u A u Ak can be written as u A ¼ k such that Ai ¼ A ¼ for all 2 i k. Let u A u Ak u A ¼ k such that Ai ¼ A ¼ for all 2 i k. Then u A u Ak ( u A ¼ u Ak )( u A ¼ 2 k ) ( u A ¼ A k u A ¼ A k )( u A ¼ 2 k ). We have A ¼ A k A ¼ A k and Ai ¼ A ¼ A k for all 2 i k and the assertion holds. Now we show that one can write u A u Ak u B u Bk such that B k B. By the above statements we can assume that u A u Ak u A ¼ k such that Ai ¼ A ¼ for all 2 i k. By induction assume that u A ¼ u 2 A ¼ k u B2 u Bk such that B k B 2. Since k i 2 B i k i 2 A¼ i A ¼, setting B A ¼ we have B k B and u A u Ak u B u Bk. Theorem 2.2. Let G be a Cohen Macaulay bipartite graph as above. Then the ideal of k-covers of G, A¾E(G) Pk A, is weakly polymatroidal. Proof. Consider the ordering on the variables corresponding to the vertices of G such that x x 2 x n y y n. Let u A u Ak lex Ú B Ú Bk be two elements in the minimal generating set of A¾E(G) Pk A such that A k A and B k B (see Lemma 2.). Then there exists t, the smallest integer such that i deg xt (u A u Ak ) deg xt (Ú B Ú Bk ). It is easy to see that for any element u A ¼ k ¾ (H P ) k with A ¼ k A ¼ we have Ai lï ¼ deg xl (u A ¼ k ) i. Thus t ¾ A i and t B i and so y t u Bi. For any j t such that j ¾ A i, we have j ¾ B i. Otherwise deg x j (Ú B Ú Bk ) i and since deg x j (u A u Ak ) i we have deg x j (u A u Ak ) deg x j (Ú B Ú Bk ) which is a contradiction by assumption that t is the smallest integer with this property. Let L l tï l ¾ A i. Then as was shown L B i. For any l such that x l y t ¾ E(G), the labeling on the vertices of G implies that l t. Moreover l ¾ A i, since Supp(u Ai ) is a minimal vertex cover of G which does not contain y t. Therefore l nï x l y t ¾ E(G) L. We have u Ai and u Bi are in H P, i.e. Supp(u Ai ) and Supp(u Bi ) are minimal vertex covers of G. Then from l nï x l y t ¾ E(G) B i, we have x l Ï l ¾ B i y l Ï l ¾ Bi c, l t x t is again a minimal vertex cover of G, equivalently Û u Bi t is in H P. Then (Ú B Ú Bk )x t y t Ú B Ú Bi Ú Bi t Ú Bi Ú Bk is in A¾E(G) Pk A. Moreover, we have (Ú B Ú Bk )x t y t lex Ú B Ú Bk, which completes the proof.

7 WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS 633 The result of the above theorem seems to be true for any weighted Cohen Macaulay bipartite graph as we checked in many examples. Next we consider some classes of weighted hypergraphs for which all ideals of k-covers are either weakly polymatroidal or componentwise weakly polymatroidal. In [7, Corollary 3.2] it is shown that any ideal I PJ a PK b is componentwise linear and [7, Remark 3.3] shows that they are not necessarily polymatroidal. Here we show that these ideals are weakly polymatroidal. Theorem 2.3. Let J and K be subsets of [n]. Let I P a J P b K K [x,, x n ]. Then I is weakly polymatroidal. Proof. Let N J K, N 2 JÒ(J K ) and N 3 K Ò(J K ). Let x l,,, x l,nl be the variables correspond to the integers in N l for l, 2, 3. Consider the ordering on the variables of R such that x, x,n x 3, x 3,n3. Let u and Ú be two monomials in G(I ) such that u lex Ú. Assume that u u u 2 u 3 and Ú Ú Ú 2 Ú 3, where u l x e l, l, x e l,n l l,n l and Ú l x e¼ l, l, x e¼ l,n l l,n l, for l, 2, 3. First let x,i be the first index such that e,i e ¼,i. There exists x l, j in Supp(Ú) with x,i x l, j, since Ú u. The element h x,i (Ú x l, j ) is in I. Let x l,i be the first index such that e l,i e ¼ l,i for l 2 or 3. Since u Ú, we have deg(u 2 ) a deg(u ) a deg(ú ) deg(ú 2 ). Then there exists x l, j in Supp(Ú) with j i. The element h x l,i (Ú x l, j ) has desired properties. Theorem 2.4. Let K, J,, J s be subsets of [n] such that s i J i J i J j [t] for all i j and K J i. Let I P a 0 K P a J P a s J s K [x,, x n ]. Then I is weakly polymatroidal. Proof. Let P K (y,, y l ) and P Ji (z,, z t, x i,,, x i,bi ) for all i. Consider the following ordering on the variables of R. y y l z z t x, x,b x s, x s,bs. Any monomial u in I can be written as f Û Û 2 u u s, where Û ¾ K [y,, y l ], Û 2 ¾ K [z,, z t ], and u i ¾ K [x i,,, x i,bi ] for i,, s. For any monomial f Û Û 2 u u s ¾ G(I ) we have deg(û ) a 0, deg(u i ) a i l, where deg(û 2 ) l. Let f Û Û 2 u u s and g Û ¼ Û¼ 2 u¼ u ¼ s be two monomials in G(I ) such that f lex g. We will denote the exponent of any variable x in f, by f (x). Let x be the first variable such that f (x) g(x). The following cases may be considered: CASE (a). Let x y i for some i l. Since deg(û ) deg(û ¼ ), then for some j i we have y j ¾ Supp(Û ¼ ). Then let h x(g y j). We have h Û ¼¼ Û¼¼ 2 u¼¼ u ¼¼ s, where deg(û ¼¼) deg(û¼ ), deg(û¼¼ 2 ) deg(û¼ 2 ) and deg(u¼¼ i ) deg(u¼ i ) for all i. Thus h ¾ I. CASE (b). Let x z t for some t k. Since f g, there exists a variable x y ¾ Supp(g), where y z j for some j t or y x i, j for some i, j. Then let

8 634 F. MOHAMMADI AND S. MORADI h x(g y). We have h Û ¼¼ Û¼¼ 2 u¼¼ u ¼¼ s. If y z j, then deg(û ¼¼) deg(û¼ ), deg(û¼¼ deg(û2 ¼ ) and deg(u¼¼ i ) deg(u¼ i ) for all i. If y x i, j, then deg(û ¼¼) deg(û¼ ), deg(û¼¼ deg(û2 ¼ ) and deg(u¼¼ i ) deg(u¼ i ) for all i. Thus h ¾ I. ) 2 ) 2 CASE (c). Let x x l,i for some t k. Since Û Û ¼ and Û 2 Û ¼ 2, we have deg(u j ) deg(u ¼ j ) for any j. Therefore there exists a j ¼ j such that g(x l, j ¼) and deg(û ¼¼) deg(û¼ ), 0. Then h x(g x l, j ¼) is in I, since h Û ¼¼ Û¼¼ 2 u¼¼ u ¼¼ s deg(û ¼¼ 2 ) deg(û¼ 2 ) and deg(u¼¼ i ) deg(u¼ i ) for all i. Example.8 shows that the ideals considered in the above theorem are not necessarily componentwise weakly polymatroidal. Theorem 2.5. Let J,, J s, K be subsets of [n] such that J i J j [n] for all i j and K [n]. Let I P a J P a s J s PK b K [x,, x n ]. Then (I d ) is weakly polymatroidal for any d. Proof. Rename the variables to z,, z k, z ¼,, z¼ k ¼, x,,, x,b, y,,, y,c,, x s,,, x s,bs, y s,,, y s,cs such that variables z j ( j k) correspond to the integers in ( s i J i) K, the variables z ¼ j ( j k ¼ ) correspond to the integers in ( s i J i) Ò K, the variables x i, j correspond to the integers in [n] missing from J i K and the variables y i, j correspond to the integers in K missing from J i. Since J i J j [n], any r ¾ [n] is missing at most one of the J i. For any l, l s the only variables which do not appear in J l are x l,,, x l,bl, y l,,, y l,cl. Consider the ordering z z k z ¼ z¼ k ¼ y, y,c y s, y s,cs x, x,b x s, x s,bs on the variables of R. For any f ¾ I we can write f Û Û 2 Ú Ú s u u s, where Û ¾ K [z,, z k ], Û 2 ¾ K [z ¼,, z¼ k ], u l ¾ K [x l,,, x l,bl ] and Ú l ¾ K [y l,,, y l,cl ]. Therefore f ¾ (I d ) if and only if deg( f ) d and () deg(u i ) deg(ú i ) d a i for i,, k, (2) deg(û 2 ) deg(u ) deg(u 2 ) deg(u s ) d b. Let f Û Û 2 u u s Ú Ú s and g Û ¼ Û¼ 2 u¼ u ¼ s Ú¼ Ús ¼ be two monomials in G(I d ) such that f lex g. The exponent of any variable x in f, is denoted by f (x). Let x be the first variable such that f (x) g(x). We are going to find a variable y x such that h x(g y) ¾ I d. The following cases may be considered: CASE (a). Let x z t for some t k. Since deg( f ) deg(g), there exists a variable y ¾ Supp(g) such that y x. Since we do not have any condition on deg(û ), the monomial h x(g y) admits conditions () and (2) which implies that h ¾ I d. CASE (b). Let x zt ¼ for some t k ¼. If there exists a variable y ¾ Supp(g), where y zk ¼ for some k t or y x l,i for some l, i, then h x(g y) admits conditions () and (2). Otherwise deg(u ¼ ) deg(u¼ s ) 0 and deg(û¼ 2 ) deg(û 2). Then for any variable k in Supp(g) with k zt ¼, the element h x(g k) has desired properties (there exists such k, otherwise g f ).

9 WEAKLY POLYMATROIDAL IDEALS WITH APPLICATIONS 635 CASE (c). Let x y l,i. If there exists a variable k ¾ Supp(g), where k y l, j for j i or k x l, j for some j, then h x(g k) has desired properties. Otherwise, deg(ú ¼ l ) deg(ú l) and deg(u ¼ l ) 0. Since deg( f ) deg(g), for some l¼ l we have u l ¼ 0 or Ú l ¼ 0. Therefore there exists a variable k ¾ Supp(g), k x l ¼,i ¼ or k y l ¼,i ¼. Then the monomial h x(g k) has desired properties. CASE (d). Let x x l,i. If x l,k ¾ Supp(g) for some k i, then h x(g x l,k ) has desired properties. Otherwise we have deg(ul ¼) deg(u l). Since deg(úl ¼) deg(ú l), we have deg(ul ¼) deg(ú¼ l ) d a l. Then there exists a variable x l ¼, j ¾ Supp(ul ¼ u ¼ s ). The element h x(g x l ¼, j) has desired properties. The above theorem improves [7, Theorem 3.]. REMARK 2.6. The I P a J P a s J s P b K P b¼ K ¼ K [x,, x n ], where J,, J s are subsets of [n] such that J i J j [n] for all i j and K, K ¼ [n]. Such ideals are not necessarily componentwise linear. For example the ideal I (x, x 2 ) (x 3, x 4 ) (x 2, x 3 ) (x, x 4 ) is an ideal as described above, but is not componentwise linear. Hence Theorem 2.5 can not be extended to the case that we add to the edges J,, J s more than one random edge. ACKNOWLEDGMENTS. This paper was written during the visit of the authors to Department of Mathematics, Universitüt Duisburg-Essen Campus Essen, 457 Essen, Germany. They would like to express their deep gratitude to Professor Jürgen Herzog for his warm hospitality, support, and guidance. References [] A. Aramova, J. Herzog and T. Hibi: Weakly stable ideals, Osaka J. Math. 34 (997), [2] A. Aramova, J. Herzog and T. Hibi: Squarefree lexsegment ideals, Math. Z. 228 (998), [3] A. Aramova, J. Herzog and T. Hibi: Ideals with stable Betti numbers, Adv. Math. 52 (2000), [4] A. Conca and J. Herzog: Castelnuovo Mumford regularity of products of ideals, Collect. Math. 54 (2003), [5] A. Conca, J. Herzog and T. Hibi: Rigid resolutions and big Betti numbers, Comment. Math. Helv. 79 (2004), [6] S. Eliahou and M. Kervaire: Minimal resolutions of some monomial ideals, J. Algebra 29 (990), 25. [7] C.A. Francisco and A. Van Tuyl: Some families of componentwise linear monomial ideals, Nagoya Math. J. 87 (2007), [8] C.A. Francisco and A. Van Tuyl: Sequentially Cohen Macaulay edge ideals, Proc. Amer. Math. Soc. 35 (2007), [9] J. Herzog and T. Hibi: The depth of powers of an ideal, J. Algebra 29 (2005), [0] J. Herzog and T. Hibi: Componentwise linear ideals, Nagoya Math. J. 53 (999), [] J. Herzog and T. Hibi: Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005),

10 636 F. MOHAMMADI AND S. MORADI [2] J. Herzog and T. Hibi: Cohen Macaulay polymatroidal ideals, European J. Combin. 27 (2006), [3] J. Herzog, T. Hibi, S. Murai and Y. Takayama: Componentwise linear ideals with minimal or maximal Betti numbers, Ark. Mat. 46 (2008), [4] J. Herzog, T. Hibi and H. Ohsugi: Unmixed bipartite graphs and sublattices of the Boolean lattices, ArXiv: [5] J. Herzog, T. Hibi and N.V. Trung: Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 20 (2007), [6] J. Herzog, T. Hibi and N.V. Trung: Vertex cover algebras of unimodular hypergraphs, Proc. Amer. Math. Soc. 37 (2009), [7] J. Herzog, T. Hibi, N.V. Trung and X. Zheng Standard graded vertex cover algebras, cycles and leaves, Trans. Amer. Math. Soc. 360 (2008), [8] J. Herzog, V. Reiner and V. Welker: Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (999), [9] J. Herzog and Y. Takayama: Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002), [20] T. Hibi: Distributive lattices, affine semigroup rings and algebras with straightening laws; in Commutative Algebra and Combinatorics (Kyoto, 985), Adv. Stud. Pure Math., North- Holland, Amsterdam, 93 09, 987. [2] M. Kokubo and T. Hibi: Weakly polymatroidal ideals, Algebra Colloq. 3 (2006), [22] L. Sharifan and M. Varbaro: Betti numbers of ideals with linear quotients, to appear in Le Matematiche. [23] A.S. Jahan and X. Zheng: Pretty clean monomial ideals and linear quotients, ArXiv: Fatemeh Mohammadi Department of Pure Mathematics Faculty of Mathematics and Computer Science Amirkabir University of Technology (Tehran Polytechnic) 424, Hafez Ave., Tehran 594 Iran Current address: Department of Pure Mathematics Ferdowsi University of Mashhad P.O. Box , Mashhad Iran fatemeh.mohammadi76@gmail.com Somayeh Moradi Department of Pure Mathematics Faculty of Mathematics and Computer Science Amirkabir University of Technology (Tehran Polytechnic) 424, Hafez Ave., Tehran 594 Iran Current address: Department of Mathematics Ilam University P.O. Box , Ilam Iran somayeh.moradi@gmail.com