# Generalization off-Graphs and Their Algebraic Aspects

Authors: Fazal Ur Rehman; Hasan Mahmood; Thai Thanh Nguyen; Muhammad Ahsan Binyamin
Journal: Journal of Mathematics
 (2023)

---

## Abstract

The notion off-graphs and f-ideals are relatively new and have been studied in many papers. In this paper, we have generalized the idea of f-graphs and f-ideals to quasi f-graphs and quasi f-ideals, respectively. We have characterized all quasi f-graphs and quasi f-ideals of degree 2 and determined all the minimal primes ideals of these ideals. Furthermore, construction of quasi f-ideals of degree 2 has been described; the formula for computing Hilbert function and Hilbert series of the polynomial ring modulo the edge ideal of the quasi f-graph has been provided.


---

## Body

## 1. Introduction

The notion off-vector has a fundamental importance in algebraic, topological, and combinatorial study of simplicial complexes and polytopes. It has been studied since the time of Leonhard Euler, for example, see [1–3]. The f-ideals, which were first introduced in [4], involve the idea of f-vectors of two important simplicial complexes. More precisely, a square-free monomial ideal I of the polynomial ring R=kx1,x2,…,xn (where k is a field) is an f-ideal if and only if the f-vector of the facet complex of I coincides with the f-vector of its Stanley-Reisner complex. These ideal were first studied in [4], where the authors gave a characterization of f-ideals of degree 2. This characterization was somehow algebraic in its nature as it required the square-free monomial ideal of R to be unmixed of height n−2. The definition of f-ideal is a blend of combinatorics, algebra, and topology. In order to characterize f-ideals for any degree, it was needed to see it through combinatorial and topological aspects too. The characterization of f-ideals for homogeneous unmixed square-free monomial ideals of any degree d was given in [5], in which combinatorial aspects were also considered. Later on, the notion of f-graphs (in [6]) and f-simplicial complex (in [7]) was introduced. These notions have been studied for it various properties in the papers [4–12].In Computational Algebraic Geometry and Commutative Algebra, the notion of Hilbert polynomial and Hilbert series are very useful and important invariants of any finitely generated standard graded algebra over some field. These encode much useful information and are the easiest way for the computation of degree and dimension of an algebraic variety defined through explicit polynomials. Theorem 6.7.2 and the Proposition 6.7.3 of [13] tell that Hilbert function, and Hilbert series of the Stanley-Reisner ring can be computed through the f-vector of non-face complex. It is clear that finding f-vector of the facet complex of some square-free monomial ideal is much simpler than computing f-vector of its non-face complex. One importance of studying f-ideals lies in the fact that Hilbert series of polynomial ring modulo any f-ideal can be computed directly by looking at the f-vector of its facet complex only. However, the class of f-ideals is not that large; in addition, f-ideals exist in R=kx1,x2,…,xn only for special n’s. So far no criterion exists in the facet ideal theory which helps us in computing Hilbert function and Hilbert series of the polynomial ring modulo any square-free monomial ideal by using the f-vector of the facet complex of I.The motivation of writing this article is to extend the class of those ideals for which the Hilbert function and the Hilbert series ofR/I can be computed through the f-vector of their facet complex. The idea is to read off the f-vector of the non-face complex of I with the help of f-vector of its facet complex. For example, consider the ideal I=x1x2,x3x4,x1x3x5,x2x4x5 in the polynomial ring R=kx1,x2,x3,x4,x5. This ideal I is f-ideal (see [10]); the common f-vector of the facet complex and the non-face complex of I is (5, 8, 2). Then, by [13], [Theorem 6.7.2], the Hilbert series of R/I is equal to 1z0/1−z0+5z1/1−z1+8z2/1−z2+2z3/1−z3. However, in the same ring, the ideal J=x1x2x4,x1x2x5,x1x4x5,x2x3x5,x3x4x5 is not f-ideal because the f-vector of its facet complex is (5, 9, 10) and the f-vector of its Stanley-Reisner complex is (5, 10, 10). Although J is not f-ideal, yet we can still express the Hilbert series of R/I in terms of the f-vector of its facet complex in the following manner: 1z0/1−z0+5z1/1−z1+9z2/1−z2+10z3/1−z3 , keeping in mind that (0, 1, 0) is the difference vector. The key point is to control this difference vector. Once we are able to control the difference of f-vectors of these two complexes, we can obviously achieve the target of expressing Hilbert series of polynomial ring modulo the ideal. It is natural to name the ideal J as “quasi f-ideal of type (0, 1, 0)”. In combinatorial perspective assuming a graph whose edge ideal is a quasi f-ideal of type a,b, then we may call naturally a quasi f-graph of type a,b, where a,b∈ℤ2. For convenience, we have included the complete list of all non-isomorphic quasi f-graphs on n≤6 vertices with type indicated as shown in Figure 1.Figure 1
Quasif-graphs up to n≤6.Before giving a systematic definition of quasif-ideals, we would like to remark that every f-ideal will turn out to be quasi f-ideal of type 0 vector; moreover, unlike f-ideals, these ideals can be found in any polynomial ring in any number of variables.This paper is organized as follows: Section2 is devoted to recalling some basic definitions to make this paper self-explanatory. In the third section, we give a systematic definition of quasi f-ideal and quasi f-simplicial complex supported with some examples. Section 4 supplies two different characterizations of equigenerated quasi f-ideals of degree 2, given in Theorems 1 and 2; the Proposition 1 answers the question that which ordered pairs of ℤ2 can be realized as the type of some quasi f-ideals. Theorem 3 gives all the minimal prime ideals of any homogeneous quasi f-ideal of degree 2. Then a construction of these ideals has been given in the Proposition 3; a formulation of Hilbert function and Hilbert series for these ideals is given in Theorems 4 and 5 respectively.


## 2. Basic Set up

This section entails basic definitions and concepts which make this article self-contained. Throughout this paper, the characterk represents a field, and R is a polynomial ring over k in n variables x1,x2,…,xn. We start with the following definition of a simplicial complex.Definition 1.
(see [14]). Let V=v1,v2,…,vn be a vertex set and Δ be a subset of PV. We say Δ a simplicial complex on V if, (i) vi∈Δ for all i∈1,2,…,n, and, (ii) subsets of every element of Δ belong to Δ.
The members ofΔ are known as faces; the dimension of a face F is one less than the cardinality of F. The maximal faces under inclusion are known as facets. It is clear that a simplicial complex can be determined by its facets. If F1,F2,…,Fr are the facets of Δ, we write Δ=F1,F2,…,Fr to say that Δ is generated by these Fi′s. The dimension of a simplicial complex Δ is defined as follows:(1)dimΔ=maxdimF|Fis facet inΔ.Remark 1.
A simplicial complex whose facets can have at most dimension 1 is actually a simple graph, where the 1-dimensional facets are termed as edges, and the 0-dimensional facets are isolated vertices. A graph is usually denoted byG with EG being the set of its edges.
We need to recall few more concepts from the literature before giving the definition of (quasi)f-ideals.Definition 2.
(see [14]). The f-vector of a d-dimensional simplicial complex Δ is an element f0,f1,…,fd∈ℤd+1, where fi=F∈Δ:dimF=i for all i∈0,1,2,…,d. The f-vector of Δ is denoted by fΔ.Definition 3.
(see [15]). (facet complex and non-face complex). Consider a square-free monomial ideal I of R=kx1,x2,…,xn with GI=m1,m2,…,mr as its unique minimal monomial system of generators. Then F1,F2,…,Fr are the facets of the facet complex of I on the vertices v1,v2,…,vn, where Fi=vj:xj divide mi, where i∈1,2,…,r. The facet complex of I is denoted by δℱI. And, the non-face complex of I is a simplicial complex on V=v1,v2,…vn such that a subset vi1,vi2,…,vik of V is a face of this non-face complex if and only if the corresponding monomial xi1xi2⋯xik does not belong to I. The non-face complex of I is also known as the Stanley-Reisner complex of I, and we denote it by δNI.Definition 4.
(see [15]). (facet ideal and non-face ideal). Consider a simplicial Δ on the vertex V=v1,v2,…,vn which is generated by the facets F1,F2,…,Fr. The facet ideal of Δ is square-free monomial ideal of R=kx1,x2,…,xn which is minimally generated by the monomials m1,m2,…,mr such that mi=∏vj∈Fixj, where i∈1,2,…,r. The facet ideal of Δ is denoted by IℱΔ. And, the non-face ideal of Δ is another square-free monomial ideal of R such that any monomial xi1xi2…xik is in the non-face ideal if and only if the corresponding subset vi1,vi2,…,vik of V does not belong to the complex Δ. The non-face ideal of Δ is also known as its Stanley-Reisner ideal, and it is written as INΔ.Definition 5.
(see [4]). [Definition 6] A square-free monomial ideal I of the polynomial ring R is said to be an f-ideal if and only if the f-vector of the facet complex of I coincides with the f-vector of its Stanley-Reisner complex, i.e. fδℱI=fδNI. A simplicial complex is said to be an f-simplicial complex if its facet ideal is an f-ideal. A 1-dimensional f-simplicial complex is termed as f-graph for obvious reason.
Let us place the definition and some examples of the central notion of this paper, i.e., quasif-ideal, in the separate section. However, before moving to the next section, we would rather recall the definition of perfect sets of R. These sets are used in characterizing f-ideals as given in [11], [Theorem 2.3].
LetSmR denote the set of all square-free monomials in R; let SmRd be the set of all square-free monomials of degree d in SmR. For a subset T⊆SmR, consider the upper shadow, ⊔T, and the lower shadow of T, ⊓T, as given below:(2)⊔T=gxi|g∈T,xidoes not divideg,1≤i≤n,⊓T=hh=gxifor someg∈Tand somexiwithxig.
If, in particular,T sits in SmRd, then ⊔T⊂SmRd+1 and ⊓T=SmRd−1. The set T is then called upper perfect if T=SmRd+1, and it is said to be lower perfect if its lower shadow is full, i.e., ⊓T⊂SmRd−1. The set T is called a perfect set if and only if it is lower perfect as well as upper perfect. In general, perfect sets can have different cardinalities; for example, every subset of SmRd containing a perfect set is again a perfect set. The smallest number among the cardinalities of perfect sets of degree d is called the n,dth perfect number, and is denoted by Nn,d. By [11], [Lemma 3.2], for a positive t and n≥4, we have the following equations:(3)Nn,2=t2−t,whenn=2t;t2,whenn=2t+1.


## 3. Quasif-Ideals: Definition and Examples

In this section, we have introduced the notion of quasif-ideals, quasi f-graphs and quasi f-simplicial complexes. Some examples are also presented.Definition 6.
Leta1,a2,…,as∈ℤs. A square-free monomial ideal I in the polynomial ring R=kx1,x2,…,xn is said to be a quasi f-ideal of type a1,a2,…,as if and only if fδNI−fδℱI=a1,a2,…,as.
IfI is quasi f-ideal of type a1,a2,…,as in the ring R=kx1,x2,…,xn, then the definition requires that both the complexes of I, the facet complex and the Stanley-Reisner complex, should be s-dimensional. It means that s≤n; in fact, s+1=maxdegu:u∈GI, where G(I) is the set of minimal generators of I. Moreover, as a consequence of Kruskal-Katona theorem, we can say that not every s-tuple of integers can be realized as type of some quasi f-ideal. However, every f-ideal is a quasi f-ideal whose type is a zero vector, and obviously any quasi f-ideal with type some non-zero vector can not be f-ideal. But we would like to mention that the class of quasi f-ideals is much more bigger than the class of f-ideals; moreover, unlike f-ideals, examples of quasi f-ideals can be found in R=kx1,x2,…,xn, for any n.
Let us now consider some examples of quasif-ideals of some types below.Example 1.
Everyf-ideal is quasi f-ideal of type 0. We would like the readers to see [4–6, 10, 11] to know more about f-ideals and f-graphs.Example 2.
The idealJ=x1x2x4,x1x2x5,x1x4x5,x2x3x5,x3x4x5 of the polynomial ring in 5 variables, which has been discussed in the introduction of this paper, is quasi f-ideal of type (0,1, 0).Example 3.
Consider the monomial idealI=〈x1x2x6,x1x2x7,x1x3x4,x1x3x5, x1x3x6,x1x3x7,x1x4x5,x1x4x6,x1x5x7,x1x6x7,x2x4x5,x2x4x7,x2x6x7,x3x4x6,x3x5x7, x2x5x6,x5x6x7〉 of degree 3 in the ring R=kx1,x2,x3,x4,x5,x6,x7. The facet complex and the non-face complex of I are(4)δℱI=v1,v2,v6,v1,v2,v7,v1,v3,v4,v1,v3,v5,v1,v3,v6,v1,v3,v7v1,v4,v5,v1,v4,v6,v1,v5,v7,v1,v6,v7,v2,v4,v5,v2,v4,v7v2,v6,v7,v3,v4,v6,v3,v5,v7,v2,v5,v6,v5,v6,v7,δNI=v1,v2,v3,v1,v2,v4,v2,v3,v4,v2,v3,v6,v1,v5,v6,v3,v5,v6v3,v4,v5,v2,v5,v7,v4,v5,v7,v4,v6,v7,v3,v6,v7,v4,v5,v6,v2,v4,v6v3,v4,v7,v1,v4,v7,v2,v3,v7,v2,v3,v5,v1,v2,v5.this implies that fδℱI=7,20,17 and fδNI=7,21,18, it means that I is a quasi f-ideal of the type (0,1,1).Definition 7.
Leta1,a2,…,as∈ℤs; let Δ be a simplicial complex on the vertex set V=v1,v2,…,vn. We say that Δ is quasi f-simplicial complex of type a1,a2,…,as if the facet ideal of Δ is quasi f-ideal of type a1,a2,…,as in the ring R=kx1,x2,…,xn. It is natural to call 1-dimensional quasi f-simplicial complex as quasi f-graph. Indeed, the type of quasi f-graph would be some ordered pair of integers.Example 4.
Everyf-graph is a quasi f-graph with type 0-vector.Example 5.
The simplicial complexΔ=v1,v2,v3,v4,v3,v5,v1,v4,v5 on V=v1,v2,v3,v4,v5 is a non-pure quasi f-simplicial complex of type (0, 1, 0) as shown in Figure 2.Figure 2
Non-pure quasif-simplicial complex of type (0, 1, 0).Example 6.
A graphG=v1,v2,v2,v3,v3,v4,v3,v5,v1,v5 on V=v1,v2,v3,v4,v5 is not a quasi f-graph as shown in Figure 3.Figure 3
The geometric realization ofG.


## 4. Quasif-Graphs and Quasi f-Ideals of Degree 2

Our main goal is to characterize all quasif-graphs. Since the definition of quasi f-graph allows us to concentrate on the algebraic aspect of this notion, therefore, throughout this paper, we will use the approach that is coming though the quasi f-ideals as the edge ideals of these graphs. Firstly, we characterize all those quasi f-ideals in the ring R=kx1,x2,…,xn whose minimal generating set is a subset of the set of all square-free monomial of degree 2 in R. We may call such ideals as equigenerated (or pure) square-free monomial ideals of degree 2. Obviously, the type of such quasi f-ideals will be ordered pair a,b∈ℤ2. However, since we are to consider only those ideals for which the facet complex and the non-face complex have the same vertex set V=v1,v2,v3,…,vn, so we are bound to consider only those ideals whose support is full, i.e. an ideal I of R=kx1,x2,…,xn for which ∪u∈GIsuppu=x1,x2,…,xn. This means that if I is quasi f-ideal of type a,b, then a must be zero in the ordered pair a,b. Thus any quasi f-ideal of degree 2 must be of the type 0,b. The following theorem characterizes all such ideals.Theorem 1.
LetI be an equigenerated square-free monomial ideal of R=kx1,x2,…,xn of degree 2, and let GI=u1,u2,…,ur be the unique minimal generating set of I. Then I is quasi f-ideal of type 0,b∈ℤ2 if and only if the following conditions hold true:(1)
htI=n−2;(2)
n2≡0mod2if b is even1mod2if b is odd;(3)
GI=1/2n2−b.Proof.
First suppose thatI is quasi f-ideal of type 0,b. This means that dimδℱI=dimδNI and fδNI−fδℱI=0,b. But as I is equigenerated square-free monomial ideal of degree 2, we have that dimδℱI=1. Also, from [13], [Corollary 6.3.5], we know that dimδNI=n−htI−1. Now the equality of dimensions of these two complexes yields that htI=n−2. Moreover, as GI=r, f1δℱI=r. Therefore, by [4], [Lemma 3.2], we have f1δNI=n2−r. The fact that I is a quasi f-ideal of the type 0,b also gives us the equation: f0δNI−f0δℱI=b, hence, n2−r−r=b or n2−2r=b. This further implies that GI=r=1/2n2−b. Note that the equation n2−2r=b also tells that the parity of n2 is same as the parity of b, and hence condition (2).
Conversely, we suppose that conditions (1), (2) and (3) are satisfied. AsI is equigenerated square-free monomial ideal of degree 2, dimδℱI=1. Condition (1) together with the fact that dimδNI=n−htI−1 implies that the complex δNI also has dimension 1. As both the complexes, δℱI and δNI, have the same vertex set V=v1,v2,…,vn, so f0δNI=f0δℱI=n. This shows that f0δNI−f0δℱI=0. The other two conditions together with [4], [Lemma 3.2] give the following:(5)f1δNI−f1δℱI=n2−GI−GI=n2−2GI=n2−212n2−b=b.
ThusI is quasi f-ideal of type 0,b.Corollary 1.
LetG be a simple graph on n vertices. Then G is a quasi f-graph of type 0,b∈ℤ2 if and only if the following conditions hold true:(1)
vertex covering number ofGisn−2.(2)
n2≡0mod2if b is even1mod2if b is odd;(3)
EG=1/2n2−b.Proof.
The proof is obvious now as the vertex covering number of a graph is the cardinality of the vertex cover of smallest cardinality.Remark 2.
While characterizingf-ideals of degree 2, we were bound to deal with only those polynomial rings in n variables for which n2 was even. But this restriction is no longer needed for the case of quasi f-ideals of degree 2, as the above theorem also considers the situation when n2 is odd. However, it imposes some restrictions on the value of b; it says that if I is quasi f-ideal of degree 2 in the polynomial ring R=kx1,x2,…,xn having type 0,b, then the parity of n2 and b must be same. So, if n=4k or n=4k+1 and b is odd, then there will not be any quasi f-ideal of degree 2 of type 0,b. Similarly for n=4k+2 or n=4k+3, there will not be any quasi f-ideal of type 0,b with even b. Moreover, since b=f1δNI−f1δℱI, [4], [Lemma 3.2] shows that b can not be greater than n2. Also, as the height of a quasi f-ideal of degree 2 has to be n−2, b≠n2.
The next theorem gives a combinatorial characterization of quasif-ideals of degree 2. This theorem involves the notion of an upper perfect set.Theorem 2.
LetI be an equigenerated square-free monomial ideal of the polynomial ring R=kx1,x2,…,xn of degree 2, and let GI be its minimal generating set. Then I is quasi f-ideal of type 0,b∈ℤ2 (where b<n2) if and only if the following conditions are satisfied:(1)
the parity ofn2is same as the parity ofb,(2)
the setGIis upper perfect withGI=1/2n2−b.Proof.
Let us first suppose thatI is quasi f-ideal of type 0,b. Condition (1) is condition (2) of Theorem 1; clearly, GI=1/2n2−b, as I is a quasi f-ideal of type 0,b. Now we only have to show that GI is upper perfect. This is equivalent to say that I contains all square-free monomial of degree 3. Indeed it is so, otherwise if there is some monomial (say) xi1xi2xi3 which does not belong to I, then the corresponding subset vi1,vi2,vi3∈δNI. This means that dimδNI≤2, which is a contradiction to the fact that dimδNI≤1.
Conversely, note that conditions (2) and (3) of Theorem1 directly follows from the assumption. We only have to show that htI=n−2. As the ideal I contains all square-free monomials of degree 3 and higher, so dimδNI≤1. The fact that b<n2 yields that δNI is 1-dimensional complex, which implies that htI=n−2. Thus I is quasi f-ideal of type 0,b. □
It will be interesting to determine the bounds on the values ofb, which is given in the proposition below.Proposition 1.
LetI be a quasi f-ideal of degree 2 and type 0,b in the polynomial ring R=kx1,x2,…,xn. Then the following holds true:(6)−n2+2≤b≤n2−2Nn,2.Proof.
AsI is quasi f-ideal of degree 2 and type 0,b, Theorem 1 tells us that GI is a perfect subset of SmR2 with GI=1/2n2−b. Since Nn,2 is the smallest cardinality of perfect sets of degree 2, this means that GI≥Nn,2.(7)⇒12n2−b≥Nn,2,⇒b≤n2−2Nn,2.
Moreover, it is clear thatGI≤n2. This means that 1/2n2−b≤n2, which gives the inequality: −n2≤b. However, if b=−n2, then GI=n2. The square-free monomial ideal of degree 2 with n2 generators has height n−1. Since the quasi f-ideal of degree 2 and type 0,b should be of height n−2, so the value b=−n2 is not acceptable. Also, as the parity of b and −n2 has to be same, the immediate acceptable value of b which is greater than −n2 would be −n2+2. The value b=−n2+2 can be realized for by any square-free monomial ideal of degree 2 whose generating set consists of n2−1 generators. Thus we have that −n2+2≤b≤n2−2Nn,2. □
Now let us talk about the associated prime ideals of these ideals. Consider a quasif-ideal I of degree 2 and type 0,b and let p be any minimal prime ideal of I. By the condition (i) of Theorem 1, we have that htp∈n−2,n−1. In the next theorem, we see that which monomial prime ideals of height n−2 and n−1 belong to AssR/I.Theorem 3.
LetI be a quasi f-ideal of degree 2 and type 0,b. Then the following statements are true:(i)
A monomial prime idealp of height n−2 belongs to AssR/I if and only if the square-free quadratic monomial xixj∉GI, where xi,xj∉p.(ii)
A monomial prime idealp of height n−1 belongs to AssR/I if and only if xixj∈GI for all j, where xi∉p.Proof.
We will use the well-known one-to-one correspondence between facets ofδNI and the minimal vertex covers of δℱI, which correspond to the minimal primes of I (as given in [15]). More precisely, F is a facet of δNI if and only if p=xi|i∉F is a minimal prime of I.Case (i): SinceI is square-free monomial ideal, the associated primes of I are precisely the minimal primes. Thus, a monomial prime ideal p of height n−2 belongs to AssR/I if and only if vi,vj is a facet of δNI, where xi,xj∉p. This is equivalent to say that xixj∉GI, because I is quasi f-ideal of type 0,b and degree 2.Case (ii): Letp be a monomial prime ideal of height n−1 such that xi∉p. Then p belongs to AssR/I if and only if vi is a facet (isolated vertex) of δNI. This is equivalent to say that the sets vi,vj do not belong to δNI for all j≠i. Equivalently, xixj∈GI for all j≠i, because GI⊂SmR2.
We now move ahead and describe how quasif-ideals of degree 2 can be constructed. In particular, we show that every ordered pair 0,b, where −n2+2≤b≤n2−2Nn,2, can be realized as type of some quasi f-ideal of degree 2.Proposition 2.
LetA be a proper subset of x1,x2,…,xn. Then for the subset PA=xixj|xi,xj∈Aorxi,xj∉A of SmR2, the ideal I=PA is a quasi f-ideal in R=kx1,x2,…,xn of degree 2 and type 0,n−n−2A2/2.Proof.
It is easy to see thatGI=WA is a perfect set of SmR2. Moreover, as WA=A2+n−A2=2A2−2nA−n+n2/2=1/2n2−n−n−2A2/2, Theorem 2 shows that I is a quasi f-ideal of the type 0,n−n−2A2/2. □Proposition 3.
Letb be an integer satisfying the inequality of Proposition 1. Then the pair 0,b can be realized as the type of some quasi f-ideal provided b and n2 have same parity.Proof.
Ifn2 is even, then either n=4t or n=4t+1. Similarly, if n2 is odd, then either n=4t+2 or n=4t+3. We consider each situation case by case.Case (i):n=4t¯. For the subset A=x1,x2,…,x2t, we claim that the ideal I=WA∪D is quasi f-ideal of type 0,b, where D is any subset of SmR2−WA such that D=1/2n2−b−WA. It is enough to show that 1/2n2−b≥WA, because only then we can form such D. Note that WA=22t2=4t2−2t. The expression 1/2n2−b assumes the least value when b takes the largest value which is b=n2−2Nn,2. Now as n=4t is even, Lemma 3.2 of [11] tells that b=4t2−24t2−2t=2t. This means that 1/2n2−b=1/24t2−2t=4t2−2t=WA. Consequently, if b<n2−2Nn,2, then we have strict inequality 1/2n2−b>WA.Case (ii):n=4t+1¯. Consider the same subset A=x1,x2,…,x2t and the same ideal I=WA∪D, where D is any subset of SmR2−WA such that D=1/2n2−b−WA. In this case, WA=2t2+2t+12=4t2. Once again the least value of the expression n2−2Nn,2 is assumed for the largest b=n2−2Nn,2. Now as n=4t+1 is odd, we have b=n2−2Nn,2=4t+12−24t2=2t. This means that 1/2n2−b=1/24t+12−2t=4t2=WA. Finally, we have 1/2n2−b>WA for each b which is strictly less then n2−2Nn,2. Similarly, taking A=x1,x2,…,x2t+1, it can be shown that 1/2n2−b−WA for the cases n=4t+2 and n=4t+3.
Now choosing any subsetD of SmR2−WA such that D=1/2n2−b−WA and setting GI=WA∪D gives us quasi f-ideal I of type 0,b.Example 7.
Forn=8, we have −26≤b≤4. In order to construct quasi f-ideal I of type (say) 0,−6, set A=x1,x2,x3,x4. Then WA=x1x2,x1x3,x1x4,x2x3,x2x4,x3x4,x5x6,x5x7,x5x8,x6x7,x6x8,x7x8. Note that WA=12. Also, as GI should be equal to 1/282−−6=17, so we can choose any subset D of SmR2−WA consisting of five elements. Let it be D=x1x6,x2x7,x2x8,x3x7,x4x7. Then I=WA∪D is a quasi f-ideal of degree 2 and type 0,−6, because GI is perfect and GI=17=1/282−−6. Note that for any even b satisfying −26≤b≤4, we can construct quasi f-ideal of type (0,b) following the same procedure.Theorem 4.
LetG be a quasi f-graph on n vertices of type 0,b. Then the Hilbert function of R/I is given by(8)HRI,z=14n2−n+2bz−n2+5n−2b,where I is the edge ideal of G.Proof.
IfG is a quasi f-graph on n vertices of type 0,b, then I is a quasi f-ideal of the type 0,b. This means that f1δNI=GI+b. Therefore, f1δNI=n,|GI|+b. By [13], [Proposition 6.7.3], the Hilbert function of R/I is given by(9)HKΔ,z=∑i=0dz−1ifi=z−10f0+z−11f1=n+z−1GI+b.
On simplification, we get(10)HRI,z=n+z−1nn−14+12b=14n2−n+2bz−n2+5n−2b.Theorem 5.
LetG be a quasi f-graph on n vertices of type 0,b. Then the Hilbert series of R/I is given by(11)FRI,z=4+4n−2z+n2−5n+4+2bz241−z2,where I is the edge ideal of G.Proof.
SupposeG is a quasi f-graph on n vertices of type 0,b. This means that the edge ideal I of G is a quasi f-ideal of the type 0,b and it implies fδNI=f1δℱI=b, which further means that fδNI=GI+b. Therefore, fδNI=n,|GI|+b. Now using [13], [Theorem 6.7.2], we have that(12)FRI,z=f−1z01−z0+f0z11−z1+f1z21−z2=1+nz1−z+|GI|+bz21−z2.and after simplification, we get(13)FRI,z=1+n−2z+1−n+GI+bz21−z2,which implies the desired formula. □


## 5. Conclusion

The ideas of quasif-ideals and quasi f-simplicial complexes are relatively new. These notions are important and worth-study. For example, by [13], [Theorem 6.7.2] and [Proposition 6.7.3], we see that for a given square-free monomial ideal I the Hilbert Series and Hilbert function of the ring R/I can be computed from the f-vector of the non-face (Stanley-Reisner) complex. However, no such result is available in the literature which can complete this task by looking at the f-vector of the facet complex of the ideal, which is relatively easy and direct to compute. Note that if I is a quasi f-ideal, then [16] one may obtain the Hilbert series and Hilbert function through the facet complex directly by using Theorem 5. The idea is to read off the f-vector of the non-face complex of I with the help of f-vector of its facet [17] complex. In this paper, we have initiated the study of quasi f-graphs and quasi f-ideals of degree 2, which, in particular, include [18] the characterization theorem as well. Many things are yet to be explored and here we mention a few open problems related to this theory below.(1)
How to characterize all quasi non-puref-simplicial complexes?(2)
How to construct nonhomogeneous quasif-simplicial complexes and quasi f-ideals?(3)
How to compute the betti numbers of the class of all quasif-graphs?



---

*Source: 7984489-2023-02-28.xml*