Abstract

The notion of -graphs and -ideals are relatively new and have been studied in many papers. In this paper, we have generalized the idea of -graphs and -ideals to quasi -graphs and quasi -ideals, respectively. We have characterized all quasi -graphs and quasi -ideals of degree 2 and determined all the minimal primes ideals of these ideals. Furthermore, construction of quasi -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 -graph has been provided.

1. Introduction

The notion of -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 [13]. The -ideals, which were first introduced in [4], involve the idea of -vectors of two important simplicial complexes. More precisely, a square-free monomial ideal of the polynomial ring (where is a field) is an -ideal if and only if the -vector of the facet complex of coincides with the -vector of its Stanley-Reisner complex. These ideal were first studied in [4], where the authors gave a characterization of -ideals of degree 2. This characterization was somehow algebraic in its nature as it required the square-free monomial ideal of to be unmixed of height . The definition of -ideal is a blend of combinatorics, algebra, and topology. In order to characterize -ideals for any degree, it was needed to see it through combinatorial and topological aspects too. The characterization of -ideals for homogeneous unmixed square-free monomial ideals of any degree was given in [5], in which combinatorial aspects were also considered. Later on, the notion of -graphs (in [6]) and -simplicial complex (in [7]) was introduced. These notions have been studied for it various properties in the papers [412].

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 -vector of non-face complex. It is clear that finding -vector of the facet complex of some square-free monomial ideal is much simpler than computing -vector of its non-face complex. One importance of studying -ideals lies in the fact that Hilbert series of polynomial ring modulo any -ideal can be computed directly by looking at the -vector of its facet complex only. However, the class of -ideals is not that large; in addition, -ideals exist in only for special ’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 -vector of the facet complex of .

The motivation of writing this article is to extend the class of those ideals for which the Hilbert function and the Hilbert series of can be computed through the -vector of their facet complex. The idea is to read off the -vector of the non-face complex of with the help of -vector of its facet complex. For example, consider the ideal in the polynomial ring . This ideal is -ideal (see [10]); the common -vector of the facet complex and the non-face complex of is (5, 8, 2). Then, by [13], [Theorem 6.7.2], the Hilbert series of is equal to . However, in the same ring, the ideal is not -ideal because the -vector of its facet complex is (5, 9, 10) and the -vector of its Stanley-Reisner complex is (5, 10, 10). Although is not -ideal, yet we can still express the Hilbert series of in terms of the -vector of its facet complex in the following manner: , 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 -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 as “quasi -ideal of type (0, 1, 0)”. In combinatorial perspective assuming a graph whose edge ideal is a quasi -ideal of type , then we may call naturally a quasi -graph of type , where . For convenience, we have included the complete list of all non-isomorphic quasi -graphs on vertices with type indicated as shown in Figure 1.

Before giving a systematic definition of quasi -ideals, we would like to remark that every -ideal will turn out to be quasi -ideal of type 0 vector; moreover, unlike -ideals, these ideals can be found in any polynomial ring in any number of variables.

This paper is organized as follows: Section 2 is devoted to recalling some basic definitions to make this paper self-explanatory. In the third section, we give a systematic definition of quasi -ideal and quasi -simplicial complex supported with some examples. Section 4 supplies two different characterizations of equigenerated quasi -ideals of degree 2, given in Theorems 1 and 2; the Proposition 1 answers the question that which ordered pairs of can be realized as the type of some quasi -ideals. Theorem 3 gives all the minimal prime ideals of any homogeneous quasi -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 character represents a field, and is a polynomial ring over in variables . We start with the following definition of a simplicial complex.

Definition 1. (see [14]). Let be a vertex set and be a subset of . We say a simplicial complex on if, (i) for all , and, (ii) subsets of every element of belong to .
The members of are known as faces; the dimension of a face is one less than the cardinality of . The maximal faces under inclusion are known as facets. It is clear that a simplicial complex can be determined by its facets. If are the facets of , we write to say that is generated by these . The dimension of a simplicial complex is defined as follows:

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 by with being the set of its edges.
We need to recall few more concepts from the literature before giving the definition of (quasi) -ideals.

Definition 2. (see [14]). The -vector of a -dimensional simplicial complex is an element , where for all . The -vector of is denoted by .

Definition 3. (see [15]). (facet complex and non-face complex). Consider a square-free monomial ideal of with as its unique minimal monomial system of generators. Then are the facets of the facet complex of on the vertices , where divide , where . The facet complex of is denoted by . And, the non-face complex of is a simplicial complex on such that a subset of is a face of this non-face complex if and only if the corresponding monomial does not belong to . The non-face complex of is also known as the Stanley-Reisner complex of , and we denote it by .

Definition 4. (see [15]). (facet ideal and non-face ideal). Consider a simplicial on the vertex which is generated by the facets . The facet ideal of is square-free monomial ideal of which is minimally generated by the monomials such that , where . The facet ideal of is denoted by . And, the non-face ideal of is another square-free monomial ideal of such that any monomial is in the non-face ideal if and only if the corresponding subset of does not belong to the complex . The non-face ideal of is also known as its Stanley-Reisner ideal, and it is written as .

Definition 5. (see [4]). [Definition 6] A square-free monomial ideal of the polynomial ring is said to be an -ideal if and only if the -vector of the facet complex of coincides with the -vector of its Stanley-Reisner complex, i.e. . A simplicial complex is said to be an -simplicial complex if its facet ideal is an -ideal. A 1-dimensional -simplicial complex is termed as -graph for obvious reason.
Let us place the definition and some examples of the central notion of this paper, i.e., quasi -ideal, in the separate section. However, before moving to the next section, we would rather recall the definition of perfect sets of . These sets are used in characterizing -ideals as given in [11], [Theorem 2.3].
Let denote the set of all square-free monomials in ; let be the set of all square-free monomials of degree in . For a subset , consider the upper shadow, , and the lower shadow of , , as given below:If, in particular, sits in , then and . The set is then called upper perfect if , and it is said to be lower perfect if its lower shadow is full, i.e., . The set 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 containing a perfect set is again a perfect set. The smallest number among the cardinalities of perfect sets of degree is called the perfect number, and is denoted by . By [11], [Lemma 3.2], for a positive and , we have the following equations:

3. Quasi -Ideals: Definition and Examples

In this section, we have introduced the notion of quasi -ideals, quasi -graphs and quasi -simplicial complexes. Some examples are also presented.

Definition 6. Let . A square-free monomial ideal in the polynomial ring is said to be a quasi -ideal of type if and only if .
If is quasi -ideal of type in the ring , then the definition requires that both the complexes of , the facet complex and the Stanley-Reisner complex, should be -dimensional. It means that ; in fact, , where G(I) is the set of minimal generators of . Moreover, as a consequence of Kruskal-Katona theorem, we can say that not every -tuple of integers can be realized as type of some quasi -ideal. However, every -ideal is a quasi -ideal whose type is a zero vector, and obviously any quasi -ideal with type some non-zero vector can not be -ideal. But we would like to mention that the class of quasi -ideals is much more bigger than the class of -ideals; moreover, unlike -ideals, examples of quasi -ideals can be found in , for any .
Let us now consider some examples of quasi -ideals of some types below.

Example 1. Every -ideal is quasi -ideal of type 0. We would like the readers to see [46, 10, 11] to know more about -ideals and -graphs.

Example 2. The ideal of the polynomial ring in 5 variables, which has been discussed in the introduction of this paper, is quasi -ideal of type (0,1, 0).

Example 3. Consider the monomial ideal , , of degree 3 in the ring . The facet complex and the non-face complex of  arethis implies that and , it means that is a quasi -ideal of the type (0,1,1).

Definition 7. Let ; let be a simplicial complex on the vertex set . We say that is quasi -simplicial complex of type if the facet ideal of is quasi -ideal of type in the ring . It is natural to call 1-dimensional quasi -simplicial complex as quasi -graph. Indeed, the type of quasi -graph would be some ordered pair of integers.

Example 4. Every -graph is a quasi -graph with type 0-vector.

Example 5. The simplicial complex on is a non-pure quasi -simplicial complex of type (0, 1, 0) as shown in Figure 2.

Example 6. A graph on is not a quasi -graph as shown in Figure 3.

4. Quasi -Graphs and Quasi -Ideals of Degree 2

Our main goal is to characterize all quasi -graphs. Since the definition of quasi -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 -ideals as the edge ideals of these graphs. Firstly, we characterize all those quasi -ideals in the ring whose minimal generating set is a subset of the set of all square-free monomial of degree 2 in . We may call such ideals as equigenerated (or pure) square-free monomial ideals of degree 2. Obviously, the type of such quasi -ideals will be ordered pair . However, since we are to consider only those ideals for which the facet complex and the non-face complex have the same vertex set , so we are bound to consider only those ideals whose support is full, i.e. an ideal of for which . This means that if is quasi -ideal of type , then must be zero in the ordered pair . Thus any quasi -ideal of degree 2 must be of the type . The following theorem characterizes all such ideals.

Theorem 1. Let be an equigenerated square-free monomial ideal of of degree 2, and let be the unique minimal generating set of . Then is quasi -ideal of type if and only if the following conditions hold true:(1);(2);(3).

Proof. First suppose that is quasi -ideal of type . This means that and . But as is equigenerated square-free monomial ideal of degree 2, we have that . Also, from [13], [Corollary 6.3.5], we know that . Now the equality of dimensions of these two complexes yields that . Moreover, as , . Therefore, by [4], [Lemma 3.2], we have . The fact that is a quasi -ideal of the type also gives us the equation: , hence, or . This further implies that . Note that the equation also tells that the parity of is same as the parity of , and hence condition (2).
Conversely, we suppose that conditions (1), (2) and (3) are satisfied. As is equigenerated square-free monomial ideal of degree 2, . Condition (1) together with the fact that implies that the complex also has dimension 1. As both the complexes, and , have the same vertex set , so . This shows that . The other two conditions together with [4], [Lemma 3.2] give the following:Thus is quasi -ideal of type .

Corollary 1. Let be a simple graph on vertices. Then is a quasi -graph of type if and only if the following conditions hold true:(1)vertex covering number ofis.(2);(3).

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 characterizing -ideals of degree 2, we were bound to deal with only those polynomial rings in variables for which was even. But this restriction is no longer needed for the case of quasi -ideals of degree 2, as the above theorem also considers the situation when is odd. However, it imposes some restrictions on the value of ; it says that if is quasi -ideal of degree 2 in the polynomial ring having type , then the parity of and must be same. So, if or and is odd, then there will not be any quasi -ideal of degree 2 of type . Similarly for or , there will not be any quasi -ideal of type with even . Moreover, since , [4], [Lemma 3.2] shows that can not be greater than . Also, as the height of a quasi -ideal of degree 2 has to be , .
The next theorem gives a combinatorial characterization of quasi -ideals of degree 2. This theorem involves the notion of an upper perfect set.

Theorem 2. Let be an equigenerated square-free monomial ideal of the polynomial ring of degree 2, and let be its minimal generating set. Then is quasi -ideal of type (where ) if and only if the following conditions are satisfied:(1)the parity ofis same as the parity of,(2)the setis upper perfect with.

Proof. Let us first suppose that is quasi -ideal of type . Condition (1) is condition (2) of Theorem 1; clearly, , as is a quasi -ideal of type . Now we only have to show that is upper perfect. This is equivalent to say that contains all square-free monomial of degree 3. Indeed it is so, otherwise if there is some monomial (say) which does not belong to , then the corresponding subset . This means that , which is a contradiction to the fact that .
Conversely, note that conditions (2) and (3) of Theorem 1 directly follows from the assumption. We only have to show that . As the ideal contains all square-free monomials of degree 3 and higher, so . The fact that yields that is 1-dimensional complex, which implies that . Thus is quasi -ideal of type .
It will be interesting to determine the bounds on the values of , which is given in the proposition below.

Proposition 1. Let be a quasi -ideal of degree 2 and type in the polynomial ring . Then the following holds true:

Proof. As is quasi -ideal of degree 2 and type , Theorem 1 tells us that is a perfect subset of with . Since is the smallest cardinality of perfect sets of degree 2, this means that .Moreover, it is clear that . This means that , which gives the inequality: . However, if , then . The square-free monomial ideal of degree 2 with generators has height . Since the quasi -ideal of degree 2 and type should be of height , so the value is not acceptable. Also, as the parity of and has to be same, the immediate acceptable value of which is greater than would be . The value can be realized for by any square-free monomial ideal of degree 2 whose generating set consists of generators. Thus we have that .
Now let us talk about the associated prime ideals of these ideals. Consider a quasi -ideal of degree 2 and type and let be any minimal prime ideal of . By the condition (i) of Theorem 1, we have that . In the next theorem, we see that which monomial prime ideals of height and belong to .

Theorem 3. Let be a quasi -ideal of degree 2 and type . Then the following statements are true:(i)A monomial prime ideal of height belongs to if and only if the square-free quadratic monomial , where .(ii)A monomial prime ideal of height belongs to if and only if for all , where .

Proof. We will use the well-known one-to-one correspondence between facets of and the minimal vertex covers of , which correspond to the minimal primes of (as given in [15]). More precisely, is a facet of if and only if is a minimal prime of .Case (i): Since is square-free monomial ideal, the associated primes of are precisely the minimal primes. Thus, a monomial prime ideal of height belongs to if and only if is a facet of , where . This is equivalent to say that , because is quasi -ideal of type and degree 2.Case (ii): Let be a monomial prime ideal of height such that . Then belongs to if and only if is a facet (isolated vertex) of . This is equivalent to say that the sets do not belong to for all . Equivalently, for all , because .We now move ahead and describe how quasi -ideals of degree 2 can be constructed. In particular, we show that every ordered pair , where , can be realized as type of some quasi -ideal of degree 2.

Proposition 2. Let be a proper subset of . Then for the subset of , the ideal is a quasi -ideal in of degree 2 and type .

Proof. It is easy to see that is a perfect set of . Moreover, as , Theorem 2 shows that is a quasi -ideal of the type .

Proposition 3. Let be an integer satisfying the inequality of Proposition 1. Then the pair can be realized as the type of some quasi -ideal provided and have same parity.

Proof. If is even, then either or . Similarly, if is odd, then either or . We consider each situation case by case.Case (i): . For the subset , we claim that the ideal is quasi -ideal of type , where is any subset of such that . It is enough to show that , because only then we can form such . Note that . The expression assumes the least value when takes the largest value which is . Now as is even, Lemma 3.2 of [11] tells that . This means that . Consequently, if , then we have strict inequality .Case (ii): . Consider the same subset and the same ideal , where is any subset of such that . In this case, . Once again the least value of the expression is assumed for the largest . Now as is odd, we have . This means that . Finally, we have for each which is strictly less then . Similarly, taking , it can be shown that for the cases and .Now choosing any subset of such that and setting gives us quasi -ideal of type .

Example 7. For , we have . In order to construct quasi -ideal of type (say) , set . Then . Note that . Also, as should be equal to , so we can choose any subset of consisting of five elements. Let it be . Then is a quasi -ideal of degree 2 and type , because is perfect and . Note that for any even satisfying , we can construct quasi -ideal of type (0,b) following the same procedure.

Theorem 4. Let be a quasi -graph on vertices of type . Then the Hilbert function of is given bywhere is the edge ideal of .

Proof. If is a quasi -graph on vertices of type , then is a quasi -ideal of the type . This means that . Therefore, . By [13], [Proposition 6.7.3], the Hilbert function of is given byOn simplification, we get

Theorem 5. Let be a quasi -graph on vertices of type . Then the Hilbert series of is given bywhere is the edge ideal of .

Proof. Suppose is a quasi -graph on vertices of type . This means that the edge ideal of is a quasi -ideal of the type and it implies , which further means that . Therefore, . Now using [13], [Theorem 6.7.2], we have thatand after simplification, we getwhich implies the desired formula. □

5. Conclusion

The ideas of quasi -ideals and quasi -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 the Hilbert Series and Hilbert function of the ring can be computed from the -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 -vector of the facet complex of the ideal, which is relatively easy and direct to compute. Note that if is a quasi -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 -vector of the non-face complex of with the help of -vector of its facet [17] complex. In this paper, we have initiated the study of quasi -graphs and quasi -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-pure -simplicial complexes?(2)How to construct nonhomogeneous quasi -simplicial complexes and quasi -ideals?(3)How to compute the betti numbers of the class of all quasi -graphs?

Data Availability

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The early version of this paper titled “Quasi ƒ-Ideals” has been published on arxiv [19]. The second and the last authors are supported by the Higher Education Commission of Pakistan for this research (Grant no. 7515).