
The hesitant fuzzy MP filter and the hesitant fuzzy congruence relation of algebra are introduced in this study, and their properties are investigated. The comparable characterization of a hesitant fuzzy MP filter is then provided. Furthermore, we established that the set of all hesitant fuzzy congruence relations and the set of all hesitant fuzzy MP filters of R0-algebra are complete lattice isomorphism based on the features of the hesitant fuzzy congruence relation in R0-algebra.
Citation: Man Jiang. Properties of R0-algebra based on hesitant fuzzy MP filters and congruence relations[J]. AIMS Mathematics, 2022, 7(7): 13410-13422. doi: 10.3934/math.2022741
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The hesitant fuzzy MP filter and the hesitant fuzzy congruence relation of algebra are introduced in this study, and their properties are investigated. The comparable characterization of a hesitant fuzzy MP filter is then provided. Furthermore, we established that the set of all hesitant fuzzy congruence relations and the set of all hesitant fuzzy MP filters of R0-algebra are complete lattice isomorphism based on the features of the hesitant fuzzy congruence relation in R0-algebra.
Let S=K[x1,…,xn] be the polynomial ring in n variables over a field K. Let M be a finitely generated Zn-graded S-module. The K subspace aK[W] which is generated by all elements of the form aw where a is a homogeneous element in M, w is a monomial in K[W] and W⊆{x1,x2,…,xn}. K[W], is called a Stanley space of dimension |W| if it is a free K[W]-module. A decomposition D of the K-vector space M as a finite direct sum of Stanley spaces D:M=⨁rj=1ajK[Wj], is called Stanley decomposition of M. Stanley depth of D is the minimum dimension of all the Stanley spaces. The quantity
sdepth(M):=max{sdepth(D)|DisaStanleydecompositionofM} |
is called the Stanley depth of M.
Depth of a finitely generated R-module M, where R is the local Noetherian ring with unique maximal ideal m:=(x1,…,xn), is the common length of all maximal M-sequences in m. For introduction to depth and Stanley depth we recommend the readers [5,9,15]. Stanley conjectured in [17] that for any Zn-graded S-module M, sdepth(M)≥depth(M). This conjecture has been studied in various special cases; see [6,12,14], this conjecture was later disproved by Duval et al. [4] in 2016, but it is still important to find classes of Zn-graded modules which satisfy the Stanley inequality. Let I⊂J⊂S be monomial ideals. Herzog et al. [10] showed that the invariant Stanley depth of J/I is combinatorial in nature. The most important thing about Stanley depth is that it shares some properties and bounds with homological invariant depth; see [1,6,16].
Let G=(VG,EG) be a graph with vertex set VG and edge set EG. A graph is called simple if it has no loops and multiple edges. Through out this paper all graphs are simple. A graph G is said to be connected if there is a path between any two vertices of G. If VG={v1,v2,…,vn} and S=K[x1,x2,…,xn], then edge ideal I(G) of the graph G is the ideal of S generated by all monomials of the form xixj such that {vi,vj}∈EG. Let n≥2. A path on n vertices say {u1,u2,…,uu} is a graph denoted by Pn such that EPn={{ui,ui+1}:1≤i≤n−1}. Let n≥3. A cycle on n vertices {u1,u2,…,un} is a graph denoted by Cn such that ECn={{ui,ui+1}:1≤i≤n−1}∪{u1,un}. A simple and connected graph T is said to be a tree if there exists a unique path between any two vertices of T. If u,v∈ VG then the distance between u and v is the length of the shortest path between u and v. The maximum distance between any two vertices of G is called diameter of G, denoted by d(G). The degree of a vertex u in a graph G is the number of edges incident on u, degree of u is denoted by deg(u). A graph with only one vertex is called a trivial graph. We denote the trivial graph by T. Any vertex with degree 1 is said to be a leaf or pendant vertex of G. Internal vertex is a vertex that is not a leaf. A tree with one internal vertex and k−1 leaves incident on it is called k-star, we denoted k−star by Sk.
The aim of this paper is to study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of firecracker graphs, some classes of caterpillar graphs and some newly defined unicyclic graphs with an arbitrary non-trivial connected graph G. We compute formulae for the values of depth that are the functions of depth of the quotient ring of the edge ideal I(G) see Theorem 3.1, Theorem 3.2, Theorem 3.3 and Theorem 3.4. As a consequence we also prove that if the Stanley's inequality holds for the quotient ring of the edge ideal I(G), then it also holds for the quotient rings of the edge ideals associated to the corona product of the graphs we considered with G. We also compute values of depth and Stanley depth and verify Stanley's inequality for the quotient ring of the edge ideals associated with some special classes of caterpillar graphs, see Theorem 4.1 and Theorem 4.2.
In this section some definitions from Graph Theory are presented. For more details we refer the readers to [7,8,18]. We also present some known results from Commutative Algebra that are frequently used in this paper. Note that by abuse of notation, xi will at times be used to denote both a vertex of a graph G and the corresponding variable of the polynomial ring S. For a given graph G, K[VG] will denote the polynomial ring whose variables are the vertices of the graph G.
Definition 2.1 ([7]). Let G1 and G2 be two graphs with order n and m respectively. The corona product of G1 and G2 denoted by G1∘G2, is the graph obtained by taking one copy of G1 and n copies of G2; and then by joining the i-th vertex of G1 to every vertex in the i-th copy of G2; see Figure 1.
Definition 2.2. Let z≥1 and k≥2 be integers and Pz be a path on z vertices u1,u2,…,uz that is, EPz={uiui+1:1≤i≤z−1} (for z=1, EPz=∅). We define a graph on zk vertices by attaching k−1 pendant vertices at each ui. We denote this graph by Pz,k; see Figure 2.
Definition 2.3. Let z≥3 and k≥2 be integers and Cz be a cycle on z vertices u1,u2,…,uz that is, ECz={uiui+1:1≤i≤z−1}∪{u1uz}. We define a graph on zk vertices by attaching k−1 pendant vertices at each ui. We denote this graph by Cz,k; see Figure 2.
Definition 2.4 ([18]). Firecracker is a graph formed by the concatenation of α number of k-stars by linking exactly one leaf from each star. It is denoted by Fα,k; see Figure 3.
Definition 2.5. The graph obtained by joining the end vertices of the path joining the leaves of the α stars in Fα,k. We call this graph circular firecracker and is denoted by CFα,k; see Figure 3.
Definition 2.6. Let z≥3 be an odd integer and k1,k3,k5,…,kz be integers greater than 1. Let Pz be a path on z vertices u1,u2,…,uz that is, EPz={uiui+1:1≤i≤z−1}. Let a∈{1,3,5,…,z}, we define a graph by attaching ka−1 pendant vertices at each vertex ua of Pz. We denote this graph by Pz; see Figure 4.
Definition 2.7. Let z≥2 and k≥3 be integers and Pz be a path on z vertices {u1,u2,…,uz} that is, EPz={uiui+1:1≤i≤z−1}. We denote by Pz,k the graph obtained by attaching k+i−2 pendant vertices at each ui of Pz; see Figure 4.
Here we recall some known results that will be used in this paper.
Lemma 2.8. ([2,Proposition 1.2.9]). (Depth Lemma) If 0⟶E1⟶E2⟶E3⟶ 0 is a short exact sequence of modules over a local ring S, or a Noetherian graded ring with S0 local then
(1) depth(E1)≥min{depth(E2),1+depth(E3)}.
(2) depth(E2)≥min{depth(E1),depth(E3)}.
(3) depth(E3)≥min{depth(E1)−1,depth(E2)}.
Lemma 2.9 ([14,Lemma 2.4]). If 0→E1→E→E2→0 is a short exact sequence of Zn−graded S-module, then
sdepth(E)≥min{sdepth(E1),sdepth(E2)}. |
Proposition 1 ([16,Corollary 1.3]). If I⊂S is a monomial ideal and u∈S is a monomial such that u∉I, then depthS(S/(I:u))≥depthS(S/I).
Proposition 2 ([3,Proposition 2.7]). If I⊂S is a monomial ideal and u∈S is monomial such that u∉I, then sdepthS(S/(I:u))≥sdepthS(S/I).
Lemma 2.10 ([13,Lemma 3.6]). Let I⊂S be a monomial ideal. If S′=S⊗KK[xn+1]≅S[xn+1], then depth(S′/I′S′)=depth(S/I)+1 and sdepth(S′/I′S′)=sdepth(S/I)+1.
Lemma 2.11 ([3,Proposition 1.1]). If I′⊂S′=K[x1,…,xm] and I″⊂S″=K[xm+1,…,xn] are monomial ideals, with 1≤m<n, then
depthS(S/(I′S+I″S))=depthS′(S′/I′)+depthS″(S″/I″). |
Lemma 2.12 ([3,Proposition 1.1]). If I′⊂S′=K[x1,…,xm] and I″⊂S″=K[xm+1,…,xn] are monomial ideals, with 1≤m<n, then
depth(S′/I′⊗KS″/I″)=depthS(S/(I′S+I″S))=depthS′(S′/I′)+depthS″(S″/I″). |
Proof. Proof follows by [19,Proposition 2.2.20] and [19,Theorem 2.2.21].
Theorem 2.1 ([16,Theorem 3.1]). If I′⊂S′=K[x1,…,xm] and I″⊂S″=K[xm+1,…,xn] are monomial ideals, with 1≤m<n, then
sdepthS(S/(I′S+I″S))≥sdepthS′(S′/I′)+sdepthS″(S″/I″). |
Lemma 2.13. If I′⊂S′=K[x1,…,xm] and I″⊂S″=K[xm+1,…,xn] are monomial ideals, with 1≤m<n, then
sdepth(S′/I′⊗KS″/I″))≥sdepthS′(S′/I′)+sdepthS″(S″/I″). |
Proof. By [19,Proposition 2.2.20], we have S′/I′⊗KS″/I″≅S/(I′S+I″S), by Theorem 2.1 the required result follows.
Let m≥2 be an integer, and consider {Mj:1≤j≤m} and {Ni:0≤i≤m} be sequence of Zn−graded S−modules and consider the chain of short exact sequences of the form
0⟶ M1⟶ N0⟶ N1⟶ 0 |
0⟶ M2⟶ N1⟶ N2⟶ 0 |
⋮
0⟶ Mm−1⟶ Nm−2⟶ Nm−1⟶ 0 |
0⟶ Mm⟶ Nm−1⟶ Nm⟶ 0. |
Then the following lemmas play key role in the proofs of our theorems.
Lemma 2.14. ([11,Lemma 3.1]). If depthMm≤depthNm and depthMj−1≤depthMj, for all 2≤j≤m, then depthM1=depthN0.
Lemma 2.15. sdepthN0≥min{sdepthMj,sdepthNm:1≤j≤m}.
Proof. Proof follows by applying Lemma 2.9 on the above chain of short exact sequences.
Proposition 3 ([1]). If I is an edge ideal of n-star, then depth(S/I)=sdepth(S/I)=1, and depth(S/It),sdepth(S/It)≥1.
Corollary 2.16 ([6,Theorem 3.2]). Let G be a connected graph. If I=I(G) and d is the diameter of G, then
depth(S/I)≥⌈d+13⌉. |
Theorem 2.2 ([6,Theorem 4.18]). Let G be a connected graph. If I=I(G) and d is the diameter of G, then for 1≤t≤3 we have
sdepth(S/It)≥⌈d−4t+53⌉. |
Corollary 2.17. Let G be connected graph. If I=I(G) and d is the diameter of G, then we have
sdepth(S/I)≥⌈d+13⌉. |
In this section we prove our main results related to corona product of graphs. We start this section with some elementary results that are necessary for our main results. Let T be a trivial graph and G any non-trivial and connected graph. The first lemma of this section give depth and Stanley depth of the cyclic modules associated with T∘G. For examples of T∘G; see Figure 5.
Lemma 3.1. Let T be a trivial graph and G be any connected non-trivial graph. If I=I(T∘G) and S:=K[V(T∘G)], then depth(S/I)=1 and sdepth(S/I)=1.
Proof. By definition of T∘G the only vertex x of T has an edge with every vertex of G. Consider the following short exact sequence
0⟶ S/(I:x)⟶ S/I⟶ S/(I,x)⟶ 0. |
Therefore S/(I:x)≅K[x], and depth(S/(I:x))=1. Now S/(I,x)≅Sx/I(G), where Sx:=S/(x). We have depth(S/(I,x))=depth(Sx/I(G))≥1, by Corollary 2.16. Now by using Depth Lemma, we have depth(S/I)=1. For the Stanley depth since S/(I:x)≅K[x] we have sdepth(S/(I:x))=1. Now S/(I,x)≅Sx/I(G). We have sdepth(S/(I,x))=sdepth(Sx/I(G))≥1, by using Lemma 2.9 and Proposition 2, we have sdepth(S/I)=1.
Proposition 4. For n,k≥2, let G be a non-trivial connected graph. If S:=K[V(Sk∘G)], then
depth(S/I(Sk∘G))=k−1+t, |
where t=depth(K[V(G)]/I(G)). Also
sdepth(S/I(Sk∘G))≥k−1+s, |
where s=sdepth(K[V(G)]/I(G)).
Proof. First we prove the result for depth. Let k=2. If e be a variable corresponding to a leaf in S2. Consider the following short exact sequence
0⟶ S/(I:e)⟶ S/I⟶ S/(I,e)⟶ 0 |
it is easy to see that S/(I:e)≅K[V(G)]/I(G)⊗KK[e] and
S/(I,e)≅K[V(T∘G)]/I(T∘G)⊗KK[V(G)]/I(G)). |
By Lemma 3.1, Lemma 2.10 and [19,Theorem 2.2.21], we have depth(S/(I:e))=1+t and depth(S/(I,e))=1+t=depth(S/(I:e)). Thus by Depth Lemma we have depth(S/I)=1+t.
Let k≥3. We will prove the required result by induction on k. Let e be a variable corresponding to a leaf in Sk. Consider the following short exact sequence
0⟶ S/(I:e)⟶ S/I⟶ S/(I,e)⟶ 0 |
we have
S/(I:e)≅k−2⊗Kj=1K[V(T∘G)]/I(T∘G)⊗KK[V(G)]/I(G)⊗KK[e]. |
By Lemmma [19,Theorem 2.2.21], we have
depth(S/(I:e))=k−2∑j=1depth(K[V(T∘G)]/I(T∘G))+depth(K[V(G)]/I(G))+depthK[e], |
by Lemma 3.1, we get depth(S/(I:e))=k−2+t+1=k−1+t. It can easily be seen that
S/(I,e)≅K[V(Sk−1∘G)]/I(Sk−1∘G)⊗K(K[V(G)]/I(G)). |
Thus by [19,Theorem 2.2.21]
depth(S/(I,e))≅depth(K[V(Sk−1∘G)]/I(Sk−1∘G))+depth((K[V(G)]/I(G))), |
applying induction on k we get
depth(S/(I,e))=(k−2+t)+t=k+2t−2≥k−1+t=depth(S/(I:e)). |
Hence by Depth Lemma we have depth(S/I)=k−1+t. This completes the proof for depth.
For Stanley depth the result follows by Lemma 2.13 instead of [19,Theorem 2.2.21] and Lemma 2.9 instead of Depth Lemma.
Corollary 3.2. If Stanley's inequality holds for K[V(G)]/I(G) then it also holds for S/I(Sk∘G).
Theorem 3.1. Let z≥1 and k≥2 be integers. If G is a connected graph with |V(G)|≥2 and S:=K[V(Pz,k∘G)], then
depth(S/I(Pz,k∘G))=z(k−1+t), |
where t=depth((K[V(G)])/I(G)) and
sdepth(S/I(Pz,k∘G))≥z(k−1+s), |
where s=depth((K[V(G)])/I(G)); see Figure 6.
Proof. First we prove the result for depth. We consider the following cases.
1.If z=1 and k≥2 then the result follows from Proposition 4.
2.Let z=2. We consider the following subcases:
(a) If k=2 and e is a variable corresponding to a leaf in P2,2. Consider the following short exact sequence
0⟶ S/(I:e)⟶ S/I⟶ S/(I,e)⟶ 0 |
then S/(I:e)≅K[V(S2∘G)]/I(S2∘G)⊗KK[V(G)]/I(G)⊗KK[e],
S/(I,e)≅K[V(S3∘G)]/I(S3∘G)⊗KK[V(G)]/I(G)). By [19,Theorem 2.2.21],
depth(S/(I:e))=depth(K[V(S2∘G)]/I(S2∘G))+depth(K[V(G)]/I(G))+depth(K[e])depth(S/(I,e))=depth(K[V(S3∘G)]/I(S3∘G))+depth(K[V(G)]/I(G))). |
By Proposition 4 we have depth(S/(I:e))=t+1+t+1=2(1+t) and depth(S/(I,e))=2+t+2=2(1+t)=depth(S/(I:e)). Hence by Depth Lemma we have depth(S/I)=2(1+t) and we are done in this special case.
(b) Let k≥3. Let e1,e2,…,ek−1 be leaves attached to u2 in P2,k and I=I(P2,k∘G). For 0≤i≤k−2, Ii:=(Ii,ei+1), where I0=I. Consider the chain of short exact sequences of the form
0⟶ S/(I0:e1)⟶ S/I0⟶ S/(I0,e1)⟶ 00⟶ S/(I1:e2)⟶ S/I1⟶ S/(I1,e2)⟶ 0⋮0⟶ S/(Ik−2:ek−1)⟶ S/Ik−2⟶ S/(Ik−2,ek−1)⟶ 00⟶ S/(Ik−1:u2)⟶ S/Ik−1⟶ S/(Ik−1,u2)⟶ 0 |
S/(Ii:ei+1)≅K[V(Sk∘G)]/I(Sk∘G)k−2−i⊗Kj=1K[V(T∘G)]/I(T∘G)i+1⊗Kj=1K[V(G)]/I(G)⊗KK[ei+1]. |
By [19,Theorem 2.2.21]
depth(S/(Ii:ei+1))=depth(K[V(Sk∘G)]/I(Sk∘G))+∑i+1j=1depth(K[V(G)]/I(G))+k−2−i∑j=1depth(K[V(T∘G)]/I(T∘G))+depth(k[ei+1]) | (3.1) |
hence by Lemma 3.1 and Proposition 4, we get
depthS/(Ii:ei+1)=k−1+t+k−2−i∑j=11+i+1∑j=1t+1=k+t+k−2−i+(i+1)t=2(k−1+t)+i(t−1). | (3.2) |
Also we have
S/(Ik−1:u2)≅k−1⊗Kj=1K[V(T∘G)]/I(T∘G)k⊗Kj=1K[V(G)]/I(G)⊗KK[u1], |
S/(Ik−1,u2)≅K[V(Sk∘G)]/I(Sk∘G)k⊗Kj=1K[V(G)]/I(G). |
By [19,Theorem 2.2.21] we have
depth(S/(Ik−1:u2))=(k−1∑j=1depth(K[V(T∘G)]/I(T∘G)))+(k∑j=1depth(K[V(G)]/I(G)))+depth(K[u1]) |
and similarly
depthS/(Ik−1,u2)=depthK[V(Sk∘G)]/I(Sk∘G)+k∑j=1depthK[V(G)]/I(G) |
by Proposition 3.1, we get
depth(S/(Ik−1:u2))=k+kt=2(k−1+t)+(k−2)(t−1), | (3.3) |
depth(S/(Ik−1,u2))=k−1+t+kt=2(k−1+t)+(k−1)(t−1). | (3.4) |
Hence by Lemma 2.14, we have
depth(S/I(P2,k∘G))=2(k−1+t). |
This completes the proof for z=2.
3. Let z≥3. We consider the following subcases:
(a) If k=2, We will prove the result by induction on z. Let uz be the vertex in the definition of Pz,2. Consider the following short exact sequence
0⟶ S/(I:uz)⟶ S/I⟶ S/(I,uz)⟶ 0 |
we have S/(I:uz)≅K[V(Pz−2,2∘G)]/I(Pz−2,2∘G)2⊗Kj=1K[V(G)]/I(G)
⊗KK[V(T∘G)]/I(T∘G)⊗KK[e],
S/(I,uz)≅K[V(Pz−1,2∘G)]/I(Pz−1,2∘G)⊗KK[V(G)]/I(G))⊗KK[V(T∘G)]/I(T∘G). |
By induction on z, [19,Theorem 2.2.21], and Lemma 3.1, we have
depth(S/(I:uz))=(z−2)(t+1)+2t+2=z(t+1) |
and similarly
depth(S/(I,uz))=z(1+t)=depth(S/(I:uz)). |
Thus by Depth Lemma we have depth(S/I)=z(1+t) and the result is proved for the case k=2.
(b) Now consider k≥3. Let e1,e2,…,ek−1 be leaves attached to uz and I=I(Pz,k∘G). For 0≤i≤k−2, Ii:=(Ii,ei+1) where I0=I. Consider the chain of short exact sequences of the form
0⟶ S/(I0:e1)⟶ S/I0⟶ S/(I0,e1)⟶ 00⟶ S/(I1:e2)⟶ S/I1⟶ S/(I1,e2)⟶ 0⋮0⟶ S/(Ik−2:ek−1)⟶ S/Ik−2⟶ S/(Ik−2,ek−1)⟶ 00⟶ S/(Ik−1:uz)⟶ S/Ik−1⟶ S/(Ik−1,uz)⟶ 0 |
we have,
S/(Ii:ei+1)≅K[V(Pz−1,k∘G)]/I(Pz−1,k∘G)k−2−i⊗Kj=1K[V(T∘G)]/I(T∘G)i+1⊗Kj=1K[V(G)]/I(G)⊗KK[ei+1]. |
By [19,Theorem 2.2.21] we have
depth(S/(Ii:ei+1))=depth(K[V(Pz−1,k∘G)]/I(Pz−1,k∘G))+i+1∑j=1depth(K[V(G)]/I(G))+k−2−i∑j=1depth(K[V(T∘G)]/I(T∘G))+1. |
Thus by Lemma 3.1, Proposition 4 and induction on z we get,
depthS/(Ii:ei+1)=(z−1)(k−1+t)+k−2−i∑j=11+i+1∑j=1t+1=(z−1)(k−1+t)+k−2−i+(i+1)t+1=z(k−1+t)+i(t−1). | (3.5) |
Also we have
S/(Ik−1:uz)≅K[V(Pz−2,k∘G)]/I(Pz−2,k∘G)k−1⊗Kj=1K[V(T∘G)]/I(T∘G)k⊗Kj=1K[V(G)]/I(G)⊗KK[uz] |
and similarly
S/(Ik−1,uz)≅K[V(Pz−1,k∘G)]/I(Pz−1,k∘G)k⊗Kj=1K[V(G)]/I(G). |
By [19,Theorem 2.2.21] and Proposition 4, we get
depth(S/(Ik−1:uz))=z(k−1+t)+(k−2)(t−1), | (3.6) |
depthS/(Ik−1,uz)=depthK[V(Pz−1,k∘G)]/I(Pz−1,k∘G)+k∑j=1depthK[V(G)]/I(G) |
depth(S/(Ik−1,uz))=(z−1)(k−1+t)+kt=z(k−1+t)+(k−1)(t−1). | (3.7) |
Hence by Lemma 2.14, we get
depth(S/I(Pz,k∘G))=z(k−1+t). |
This completes the proof.
For Stanley depth the result follows by Lemma 2.13 instead of [19,Theorem 2.2.21] and Lemma 2.15 instead of Lemma 2.14.
Corollary 3.3. If Stanley's inequality holds for K[V(G)]/I(G) then it also holds for S/I(Pz,k∘G).
Theorem 3.2. Let z≥3 and k≥2 be integers and G be a connected graph with |V(G)|≥2. Consider S:=K[V(Cz,k∘G)]. We have
depth(S/I(Cz,k∘G))=z(k−1+t), |
where t=depth((K[V(G)])/I(G)) and
sdepth(S/I(Cz,k∘G))≥z(k−1+s), |
where s=sdepth((K[V(G)])/I(G)); see Figure 7.
Proof. First we prove the result for depth.
1.Let z=3. We consider the following subcases:
(a) Let k=2. Let u be a variable corresponding to a vertex of C3 in C3,2. Consider the following short exact sequence
0⟶ S/(I:u)⟶ S/I⟶ S/(I,u)⟶ 0 |
we have
S/(I:u)≅2⊗Kj=1K[V(T∘G)]/I(T∘G)3⊗Kj=1K[V(G)]/I(G)⊗KK[e], |
S/(I,u)≅K[V(P2,2∘G)]/I(P2,2∘G)⊗KK[V(G)]/I(G)⊗KK[V(T∘G)]/I(T∘G). |
Hence by using Lemma 3.1, [19,Theorem 2.2.21] and Theorem 3.1, we have
depth(S/(I:u))=2+3t+1=3(t+1), |
depth(S/(I,u))=3(1+t)=depth(S/(I:e)). |
Thus by Depth Lemma we have depth(S/I)=3(1+t).
(b) Let k≥3. Let e1,e2,…,ek−1 be leaves attached to u3 in C3,k and I=I(C3,k∘G). For 0≤i≤k−2, Ii:=(Ii,ei+1) where I0=I. Consider the chain of short exact sequences of the form
0⟶ S/(I0:e1)⟶ S/I0⟶ S/(I0,e1)⟶ 00⟶ S/(I1:e2)⟶ S/I1⟶ S/(I1,e2)⟶ 0⋮0⟶ S/(Ik−2:ek−1)⟶ S/Ik−2⟶ S/(Ik−2,ek−1)⟶ 00⟶ S/(Ik−1:u3)⟶ S/Ik−1⟶ S/(Ik−1,u3)⟶ 0 |
we have,
S/(Ii:ei+1)≅K[V(P2,k∘G)]/I(P2,k∘G)k−2−i⊗Kj=1K[V(T∘G)]/I(T∘G)i+1⊗Kj=1K[V(G)]/I(G)⊗KKK[ei+1]. | (3.8) |
By using [19,Theorem 2.2.21]
depth(S/(Ii:ei+1))=depth(K[V(Pk,2∘G)]/I(Pk,2∘G))+i+1∑j=1depth(K[V(G)]/I(G))+k−2−i∑j=1depth(K[V(T∘G)]/I(T∘G))+depthK[ei+1] |
hence by Lemma 3.1, Proposition 4 and Theorem 3.3, we get
depthS/(Ii:ei+1)=2(k−1+t)+k−2−i∑j=11+i+1∑j=1+1=2(k−1+t)+k−2−i+it+t+1=3(k−1+t)+i(t−1). | (3.9) |
S/(Ik−1:u3)≅k−1⊗Kj=1K[V(T∘G)]/I(T∘G)k−1⊗Kj=1K[V(T∘G)]/I(T∘G)k+1⊗Kj=1K[V(G)]/I(G)⊗KK[u3], |
S/(Ik−1,u3)≅K[V(P2,k∘G)]/I(P2,k∘G)k⊗Kj=1K[V(G)]/I(G). |
By [19,Theorem 2.2.21]
depthS/(Ik−1,u3)=depthK[V(P2,k∘G)]/I(P2,k∘G)+k∑j=1depthK[V(G)]/I(G) |
by Lemma 3.1 and Theorem 3.3, we get
depth(S/(Ik−1:u1))=3(k−1+t)+(k−2)(t−1). | (3.10) |
depth(S/(Ik−1,u3))=2(k−1+t)+kt=3(k−1+t)+(k−1)(t−1). | (3.11) |
Hence by Lemma 2.14, we get
depth(S/I(C3,k∘G))=3(k−1+t). |
2. Let z≥4. We consider the following subcases:
(a) Let k=2. Let u be a variable corresponding to the vertex of Cz in Cz,2. Consider the following short exact sequence
0⟶ S/(I:u)⟶ S/I⟶ S/(I,u)⟶ 0 |
we have S/(I:u)≅K[V(Pz−3,2∘G)]/I(Pz−3,2∘G)3⊗Kj=1K[V(G)]/I(G)
2⊗Kj=1K[V(T∘G)]/I(T∘G)⊗KK[e],
S/(I,u)≅K[V(Pz−1,2∘G)]/I(Pz−1,2∘G)⊗KK[V(G)]/I(G))⊗KK[V(T∘G)]/I(T∘G). |
Hence by using Lemma 3.1, [19,Theorem 2.2.21] and Theorem 3.1, we have
depth(S/(I:u))=(z−3)(t+1)+3t+2+1=z(t+1) |
depth(S/(I,u))=z(1+t)=depth(S/(I:u)). |
Thus by Depth Lemma we have depth(S/I)=z(1+t).
(b) Let k≥3. Let e1,e2,…,ek−1 be leaves attached to uz in Cz,k and I=I(Cz,k∘G). For 0≤i≤k−2, Ii:=(Ii,ei+1) where I0=I. Consider the chain of short exact sequences of the form
0⟶ S/(I0:e1)⟶ S/I0⟶ S/(I0,e1)⟶ 00⟶ S/(I1:e2)⟶ S/I1⟶ S/(I1,e2)⟶ 0⋮0⟶ S/(Ik−2:ek−1)⟶ S/Ik−2⟶ S/(Ik−2,ek−1)⟶ 00⟶ S/(Ik−1:uz)⟶ S/Ik−1⟶ S/(Ik−1,uz)⟶ 0 |
S/(Ii:ei+1)≅K[V(Pz−1,k∘G)]/I(Pz−1,k∘G)k−2−i⊗Kj=1K[V(T∘G)]/I(T∘G)i+1⊗Kj=1K[V(G)]/I(G)⊗KK[ei+1]. |
By using [19,Theorem 2.2.21]
depth(S/(Ii:ei+1))=depth(K[V(Pz−1,k∘G)]/I(Pz−1,k∘G))+∑i+1j=1depth(K[V(G)]/I(G))+k−2−i∑j=1depth(K[V(T∘G)]/I(T∘G))+depthK[ei+1] | (3.12) |
by Lemma 3.1, Proposition 4 and Theorem 3.1, we get
depthS/(Ii:ei+1)=(z−1)(k−1+t)+k−2−i∑j=11+i+1∑j=1t+1=(z−1)(k−1+t)+k−2−i+it+t+1=z(k−1+t)+i(t−1). | (3.13) |
/(Ik−1:uz)≅K[V(Pz−3,k∘G)]/I(Pz−3,k∘G)k−1⊗Kj=1K[V(T∘G)]/I(T∘G)k−1⊗Kj=1K[V(T)∘G]/I(T∘G)k+1⊗Kj=1K[V(G)]/I(G)⊗KK[uz], |
S/(Ik−1,uz)≅K[V(Pz−1,k∘G)]/I(Pz−1,k∘G)k⊗Kj=1K[V(G)]/I(G) |
by Lemma 3.1, [19,Theorem 2.2.21] and Theorem 3.1, we have
depth(S/(Ik−1:uz))=z(k−1+t)+(k−2)(t−1). | (3.14) |
depth(S/(Ik−1,uz))=(z−1)(k−1+t)+kt=z(k−1+t)+(k−1)(t−1). | (3.15) |
Hence by Lemma 2.14, we will have the required result
depth(S/I(Cz,k∘G))=z(k−1+t). |
For Stanley depth the result follows by Lemma 2.13 instead of [19,Theorem 2.2.21] and Lemma 2.15 instead of Lemma 2.14.
Corollary 3.4. Stanley's inequality holds for S/I(Cz,k∘G) if it holds for K[V(G)]/I(G).
Theorem 3.3. Let α≥2 and k≥3 be integers and G be a connected graph with |V(G)|≥2 and S:=K[V(Fα,k∘G)]. Then
depthS/I(Fα,k∘G)=α(k−1+t)+⌈α−12⌉(t−1), |
where t=depth((K[V(G)])/I(G)) and
sdepthS/I(Fα,k∘G)≥α(k−1+s)+⌈α−12⌉(s−1), |
where s=sdepth((K[V(G)])/I(G)) and ⌈α⌉={n∈Z:n≥α}; see Figure 8.
Proof. We consider the following cases:
1.Let α=2. Let e1,e2,…,ek−1 be leaves attached to u2 in F(2,k) and I=I(F2,k∘G). Consider the short exact sequence of the form
0⟶ S/(I:e1)⟶ S/I⟶ S/(I,e1)⟶ 0 |
where e1 is leave of second star that is attached to the previous star.
S/(I:e1)≅K[V(Sk−1∘G)]/I(Sk−1∘G)k−2⊗Kj=1K[V(T∘G)]/I(T∘G)2⊗Kj=1K[V(G)]/I(G)⊗KK[e1], |
S/(I,e1)≅K[V(Sk∘G)]/I(Sk∘G)⊗KK[V(Sk−1∘G)]/I(Sk−1)⊗KK[V(G)]/I(G). |
By [19,Theorem 2.2.21]
depth(S/(I:e1))=depth(K[V(Sk−1∘G)]/I(Sk−1∘G))+2depth(K[V(G)]/I(G))+k−2∑j=1depth(K[V(T∘G)]/I(T∘G))+K[e1] |
hence by Lemma 3.1 and Proposition 4, we get
depthS/(I:e1)=k−2+t+k−2∑j=11+2t+1=k−2+t+k−2+2t+1=2(k−1+t)+(t−1) |
and similarly
depthS/(I,e1)=(k−1+t)+(k−2+t)+t=2(k−1+t)+(t−1) |
So by using Depth Lemma, we have
depthS/I(F2,k∘G)=2(k−1+t)+(t−1). |
2.Let α≥3. Let e1.e2,...,ek−1 be leaves attached to uα in F(z,k) and I=I(Fα,k∘G).
Consider the short exact sequence of the form
0⟶ S/(I:e1)⟶ S/I⟶ S/(I,e1)⟶ 0 |
where e1 is leave of last star that is attached to the previous star in Fα,k. We have
S/(I:e1)≅K[V(Fα−2,k∘G)]/I(Fα−2,k∘G)⊗KK[V(Sk−1∘G)]/I(Sk−1∘G)k−2⊗Kj=1K[V(T∘G)]/I(T∘G)2⊗Kj=1K[V(G)]/I(G)⊗KK[e1], |
S/(I,e1)≅K[V(Fα−1,k∘G)]/I(Fα−1,k∘G)⊗KK[V(Sk−1∘G)]/I(Sk−1∘G)⊗KK[V(G)]/I(G). |
By [19,Theorem 2.2.21]
depth(S/(I:e1))=depthK[V(Fα−2,k∘G)]/I(Fα−2,k∘G)+depth(K[V(Sk−1∘G)]/I(Sk−1∘G))+k−2∑j=1depth(K[V(T∘G)]/I(T∘G))+2depth(K[V(G)]/I(G))+depthK[e1] |
hence by Lemma 3.1, Proposition 4 and induction on α, we get
depthS/(I:e1)=(α−2)(k−1+t)+⌈α−32⌉(t−1)+(k−2+t)+k−2∑j=11+2t+1=α(k−1+t)+⌈α−12⌉(t−1) |
and similarly
depthS/(I,e1)=(α−1)(k−1+t)+⌈α−22⌉(t−1)+(k−2+t)+tdepthS/(I:e1)=α(k−1+t)+⌈α2⌉(t−1). | (3.16) |
So by using Depth Lemma, we have
depthS/(Fα,k∘G)=α(k−1+t)+⌈α−12⌉(t−1). |
For Stanley depth the result follows by Lemma 2.13 instead of [19,Theorem 2.2.21] and Lemma 2.9 instead of Depth Lemma.
Corollary 3.5. Stanley's inequality holds for S/I(Fα,k∘G) if it holds for K[V(G)]/I(G).
Theorem 3.4. Let α≥3 and k≥3 be integers and G be a connected graph with |V(G)|≥2. Consider S:=K[V(CFα,k∘G)]. Then
depthS/I(CFα,k∘G)=α(k−1+t)+⌈α2⌉(t−1), |
where t=depth((K[V(G)])/I(G)) and
sdepthS/I(Cα,k∘G)≥α(k−1+s)+⌈α2⌉(s−1), |
where s=sdepth((K[V(G)])/I(G)); see Figure 9.
Proof. We consider the following cases:
1.Consider α=3. Let e1,e2,...,ek−1 be leaves attached to u3 in CF(3,k) and I=I(CF3,k∘G).
Consider the short exact sequence of the form
0⟶ S/(I:e1)⟶ S/I⟶ S/(I,e1)⟶ 0 |
where e1 is leave of third star that is attached to the previous star and first star in CF3,k. We have
S/(I:e1)≅2⊗Kj=1K[V(Sk−1∘G)]/I(Sk−1∘G)k−2⊗Kj=1K[(T∘G)]/I(T∘G)3⊗Kj=1K[V(G)]/I(G)⊗KK[e1], |
S/(I,e1)≅K[V(F2,k∘G)]/I(F2,k∘G)⊗KK[V(Sk−1∘G)]/I(Sk−1∘G)⊗KK[V(G)]/I(G). |
By [19,Theorem 2.2.21]
depth(S/(I:e1))=2depth(K[V(Sk−1∘G)]/I(Sk−1∘G))+3depth(K[V(G)]/I(G))k−2∑j=1depth(K[V(T∘G)]/I(T∘G))+depthK[e1] |
hence by Lemma 3.1 and Proposition 4, we get
depthS/(I:e1)=2(k−2+t)+k−2∑j=11+3t+1=3(k−1+t)+2(t−1) | (3.17) |
and similarly
depthS/(I,e1)=2(k−1+t)+(t−1)+(k−2+t)+tdepthS/(I,e1)=3(k−1+t)+2(t−1). | (3.18) |
So by using Depth Lemma 2.8, we have
depthS/(CF3,k∘G)=3(k−1+t)+2(t−1). |
2.Let α≥3. Let e1,e2,…,ek−1 be leaves attached to uα in CF(α,k) and I=I(CFα,k∘G). Consider the short exact sequence of the form
0⟶ S/(I:e1)⟶ S/I⟶ S/(I,e1)⟶ 0 |
where e1 is leave of last star that is attached to the previous star and first star in CFα,k. We have
S/(I:e1)≅K[V(Fα−3,k∘G)]/I(Fα−3,k∘G)2⊗Kj=1K[V(Sk−1∘G)]/I(Sk−1∘G)k−2⊗Kj=1K[V(T∘G)]/I(T∘G)3⊗Kj=1K[V(G)]/I(G)⊗KK[e1], |
S/(I,e1)≅K[V(Fα−1,k∘G)]/I(Fα−1,k∘G)⊗KK[V(Sk−1∘G)]/I(Sk−1∘G)⊗KK[V(G)]/I(G). |
By using [19,Theorem 2.2.21]
depth(S/(I:e1))=depthK[V(Fα−3,k∘G)]/I(Fα−3,k∘G)+k−2∑j=1depth(K[V(T∘G)]/I(T∘G))+2depth(K[V(Sk−1∘G)]/I(Sk−1∘G))+3depth(K[V(G)]/I(G))+1 |
hence by Lemma 3.1, Proposition 4 and Theorem 3.3, we get
depthS/(I:e1)=(α−3)(k−1+t)+⌈α−42⌉(t−1)+2(k−2+t)+k−2∑j=11+3t+1=α(k−1+t)+⌈α2⌉(t−1) | (3.19) |
and similarly
depthS/(I,e1)=(α−1)(k−1+t)+⌈α−22⌉(t−1)+(k−2+t)+tdepthS/(I,e1)=α(k−1+t)+⌈α2⌉(t−1). | (3.20) |
So by using Depth Lemma 2.8, we have
depthS/(CFα,k∘G)=α(k−1+t)+⌈α2⌉(t−1). |
For Stanley depth the result follows by Lemma 2.13 instead of [19,Theorem 2.2.21] and Lemma 2.9 instead of Depth Lemma.
Corollary 3.6. Stanley's inequality holds for S/I(CFα,k∘G) if it holds for K[V(G)]/I(G).
In this section we calculate values of depth and Stanley depth of the quotient rings associated with edge ideals of some class of caterpillar graphs. We also prove that the values of both depth and Stanley depth for these classes of graphs are exactly the same. As a consequence the Stanley's inequality holds for the quotient ring of edge ideals of these classes of graphs.
Theorem 4.1. Let z≥3 and S=K[V(Pz)]. For a∈{1,3,5,…,z}, if ka>1 and I=I(Pz), then
depth(S/I)=sdepth(S/I)=z+12. |
Proof. The proof is done by induction on z. Let z=3. Consider the following short exact sequence
0⟶ S/(I:u3)⟶ S/I⟶ S/(I,u3)⟶ 0 |
We have (I:u3)=(x:x∈N(u3))+I(Sk1) and S/(I:u3)≅K[V(Sk1)∪{u3}]/I(Sk1), thus by Lemma 2.10 and Proposition 3, depthS/(I:u3)=sdepthS/(I:u3)=1+1=2. Clearly (I,u3)=(I(Sk1+1),u3) and S/(I,u3)≅K[V(Sk1+1)∪{u3}∪{e1,e2,…,ek3}]/I(Sk1+1) by Lemma 2.3 and Proposition 3, depthS/(I,u3)=sdepthS/(I,u3)=1+k3−1=k3, by using Depth Lemma, Lemma 2.9 and Proposition 2 we have
depthS/I(P3)=sdepthS/I(P3)=2. |
Now assume that z≥5, consider a short exact sequence of the form
0⟶ S/(I:uz)⟶ S/I⟶ S/(I,uz)⟶ 0 |
it is easy to see that (I:uz)=(x:x∈N(uz))+I(Pz−2) and S/(I:uz)≅K[V(Pz−2)∪{uz}]/I(Pz−2) so by Lemma 2.3 and induction on z, we get
depthS/(I:uz)=sdepthS/(I:uz)=z−2+12+1=z+12. |
Since (I,uz)=(I(Pz−2),uz) and
S/(I,uz)≅K[V(Pz−2)∪{uz}∪{e1,e2,…,ekz}]/(I(Pz−2),uz), |
therefore by using Lemma 2.10 and induction on z, we get
depthS/(I,uz)=sdepthS/(I,uz)=z−2+12+kz−1=z+12+kz−2. |
Hence by Depth Lemma we have depthS/I=z+12 and by Lemma 2.9 sdepthS/I≥z+12. Now for the upper bound by Proposition 2 we have sdepthS/(I)≤sdepthS/(I:uz)=z+12 and hence sdepth(S/I)=z+12.
Theorem 4.2. Let z≥2, k≥3 and S:=K[V(Pz,k)]. If I=I(Pz,k), then
depth(S/I)=sdepth(S/I)={k,if z = 2 ;⌊z2⌋(k−2)+z+∑⌈z2⌉−1m=1(z−2m),if z≥3. |
Where ⌊α⌋={n∈Z:n≤α}.
Proof. The proof is done by induction on z. Let z=2. Consider the following short exact sequence
0⟶ S/(I:u2)⋅u2→ S/I⟶ S/(I,u2)⟶ 0 |
we have (I:u2)=(x:x∈N(u2)) and S/(I:u2)≅K[L(u1)∪{u2}], where N(u2) are the neighbours of u2 and L(u1) represent the number of leaves at u1. Thus by Lemma 2.10, depth(S/(I:u2))=1+k−1=k. Also (I,u2)=(I(Sk),u2) and S/(I,u2)≅K[V(Sk)∪L(u2)]/I(Sk), therefore by Proposition 2.10, depth(S/(I,u2))=1+(k+1−1)=k+1 thus by Depth Lemma depth(S/I)=k. Now by Lemma 2.9 sdepth(S/I)≥k and by using Proposition 2 and Lemma 3 we have sdepth(S/I)≤k. Thus sdepth(S/I)=k. Let z=3. Consider the following short exact sequence
0⟶ S/(I:u3)⋅u3→ S/I⟶ S/(I,u3)⟶ 0 |
we have (I:u3)=I(Sk)+(x:x∈N(u3)) and S/(I:u3)≅[V(Sk)∪L(u2)∪{u3}]/I(Sk). Thus by Lemma 2.10 and Proposition, 3, depth(S/(I:u3))=sdepth(S/(I:u3))=1+(k+1−1)+1=k+2. Further (I,u3)=(I(P2,k),u3) and S/(I,u3)≅K[V(P2,k)∪L(u3)]/I(P2,k). Therefore by Lemma 2.10, and the above case we have depth(S/(I,u3))=sdepth(S/(I,u3))=k+(k+2−1)=2k+1. Applying Depth Lemma we get depth(S/I)=k+2. Now by Lemma 2.9 and Proposition 2 we get sdepth(S/I)=k+2. Let z≥4. Consider the following short exact sequence
0⟶ S/(I:uz)⋅uz→ S/I⟶ S/(I,uz)⟶ 0 |
it is easy to see that (I:uz)=(x:x∈N(uz))+I(Pz−2,k) and S/(I:uz)≅K[V(Pz−2,k)∪L(uz−1)∪{uz}]/I(Pz−2,k) also (I,uz)=(I(Pz−1,k),uz) and S/(I,uz)≅K[V(Pz−1,k)∪L(uz)]/I(Pz−1,k).
Thus by using induction on z and Lemma 2.10
depth(S/(I:uz))=depthK[V(Pz−2,k)]/I(Pz−2,k)+|L(uz−1)|+1=⌊z−22⌋(k−2)+(z−2)+⌈z−22⌉−1∑m=1(z−2−2m)+(k+z−3)+1=⌊z−22⌋(k−2)+z−2+⌈z−22⌉−1∑m=1(z−2−2m)+k+z−2=⌊z2⌋(k−2)−(k−2)+⌈z−22⌉−1∑m=0(z−2−2m)+k+z−2=⌊z2⌋(k−2)+z+⌈z−22⌉−1∑m=0(z−2−2m) |
introducing the transformation j:=m+1 we get depth(S/(I:uz))=⌊z2⌋(k−2)+z+∑⌈z2⌉−1j=1(z−2j), where j is dummy variable so by replacing j with m we get
depth(S/(I:uz))=⌊z2⌋(k−2)+z+⌈z2⌉−1∑m=1(z−2m). |
Now by considering the inequality ⌈x+y⌉≥⌈x⌉+⌈y⌉−1, we get
depth(S/(I,uz))=depthK[V(Pz−1,k)/I(Pz−1,k))+|L(uz)|=⌊z−12⌋(k−2)+z−1+⌈z−12⌉−1∑m=1(z−1−2m)+k+z−2≥⌊z−22⌋(k−2)+⌈z−1+2−22⌉−1∑m=0(z−1−2m)+k+z−2≥⌊z2⌋(k−2)−(k−2)+⌈z2⌉−2∑m=0(z−1−2m)+k+z−2=⌊z2⌋(k−2)+z+⌈z2⌉−2∑m=0(z−1−2m)≥⌊z2⌋(k−2)+z+⌈z2⌉−1∑m=1(z−2m). |
Thus by Depth Lemma
depth(S/I)=⌊z2⌋k+⌈z2⌉−1∑m=1(z−2m)+z−2⌊z2⌋. |
For Stanley depth the result follows by Lemma 2.9 and 2 instead of Depth Lemma. Clearly, one can see that Stanley's inequality holds for these classes of graphs.
All authors declare no conflicts of interest in this paper.
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