aa r X i v : . [ m a t h . A C ] S e p QUASI f -IDEALS ∗ HASAN MAHMOOD , FAZAL UR REHMAN , THAI THANH NGUYEN , MUHAMMADAHSAN BINYAMIN Abstract.
The notion of f -ideals is recent and has been studied in the papers[1] [2], [5], [10], [11], [12], [13], [14] and [15]. In this paper, we have generalizedthe idea of f -ideals to quasi f -ideals. This extended class of ideals is much biggerthan the class of all f -ideals. Apart from giving various characterizations of quasi f -ideals of degree 2, we have determined all the minimal primes ideals of theseideals. Moreover, construction of quasi f -ideals of degree 2 has been described;the formula for computing Hilbert function and Hilbert series of the polynomialring modulo quasi f -ideal has been provided. Key words: f -vector; facet complex; Stanley-Reisner complex; quasi f -ideal; : 13F20, 05E45, 13F55, 13C14. Introduction
The notion of f -vector has a fundamental importance in algebraic, topological,and combinatorial study of simplicial complexes and polytopes. It has been stud-ied since the time of Leonhard Euler, for example, see [3], [4] and [8]. The f -ideals,which were first introduced in [1], involve the idea of f -vectors of two important sim-plicial complexes. More precisely, a square-free monomial ideal I of the polynomialring R = k [ x , x , ..., x n ] (where k is a field) is an f -ideal if and only if the f -vectorof the facet complex of I coincides with the f -vector of its Stanley-Reisner complex.These ideal were first studied in [1], where the authors gave a characterization of f -ideals of degree 2. This characterization was somehow algebraic in its nature asit required the square-free monomial ideal of R to be unmixed of height n −
2. Thedefinition of f -ideal is a blend of combinatorics, algebra, and topology. In orderto characterize f -ideals for any degree, it was needed to see it through combinato-rial and topological aspects too. The characterization of f -ideals for homogeneousunmixed square-free monomial ideals of any degree d was given in [2], in which com-binatorial aspects were also considered. Later on, the notion of f -graphs (in [13]) ∗ The first and the last authors are supported by the Higher Education Commission of Pakistanfor this research (Grant no. 7515). Government College University Lahore, Pakistan. Government College University Faisalabad,Pakistan. Tulane University, USA; Hue University, College of Education, Vietnam.
E-mails : [email protected], [email protected], [email protected],[email protected] . nd f -simplicial complex (in [14]) was introduced. These notions have been stud-ied for it various properties in the papers [1], [2], [5], [10], [11], [12], [13], [14] and [15].In Computational Algebraic Geometry and Commutative Algebra, the notion ofHilbert polynomial and Hilbert series are very useful and important invariants ofany finitely generated standard graded algebra over some field. These encode muchuseful information and are the easiest way for the computation of degree and di-mension of an algebraic variety defined through explicit polynomials. The Theorem6.7.2 and the proposition 6.7.3 of [16] tell that Hilbert function, and Hilbert series ofthe 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 monomialideal is much simpler than computing f -vector of its non-face complex. One im-portance of studying f -ideals lies in the fact that Hilbert series of polynomial ringmodulo any f -ideal can be computed directly by looking at the f -vector of its facetcomplex only. However, the class of f -ideals is not that large; in addition, f -idealsexist in R = k [ x , x , ..., x n ] only for special n ’s. So far no criterion exists in thefacet ideal theory which helps us in computing Hilbert function and Hilbert seriesof the polynomial ring modulo any square-free monomial ideal by using the f -vectorof the facet complex of I .The motivation of writing this article is to extend the class of those ideals forwhich the Hilbert function and the Hilbert series of R/I can be computed throughthe f -vector of their facet complex. The idea is to read off the f -vector of thenon-face complex of I with the help of f -vector of its facet complex. For ex-ample, consider the ideal I = h x x , x x , x x x , x x x i in the polynomial ring R = k [ x , x , x , x , x ]. This ideal I is f -ideal (see [11]); the common f -vector ofthe facet complex and the non-face complex of I is (5 , , R/I is equal to z (1 − z ) + z (1 − z ) + z (1 − z ) + z (1 − z ) . How-ever, in the same ring, the ideal J = h x x x , x x x , x x x , x x x , x x x i is not f -ideal because the f -vector of its facet complex is (5 , ,
10) and the f -vector of itsStanley-Reisner complex is (5 , , J is not f -ideal, yet we can stillexpress the Hilbert series of R/J in terms of the f -vector of its facet complex in thefollowing manner: z (1 − z ) + (5+0) z (1 − z ) + (9+1) z (1 − z ) + (10+0) z (1 − z ) , keeping in mind that (0 , ,
0) isthe difference vector. The key point is to control this difference vector. Once we areable to control the difference of f -vectors of these two complexes, we can obviouslyachieve 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 , , f -ideals, we would like to remark that every f -idealwill turn out to be quasi f -ideal of type 0 vector; moreover, unlike f -ideals, theseideals 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 basicdefinitions to make this paper self-explanatory. In the third section, we give asystematic definition of quasi f -ideal supported with some examples. The section 4 upplies two different characterizations of equigenerated quasi f -ideals of degree 2,given in Theorem 4.1 and Theorem 4.3; the proposition 4.4 answers the question thatwhich ordered pairs of Z can be realized as the type of some quasi f -ideals. TheTheorem 4.5 gives all the minimal prime ideals of any homogeneous quasi f -idealof degree 2. Then a construction of these ideals has been given in the proposition4.7; a formulation of Hilbert function and Hilbert series for these ideals is given inTheorem 4.9 and Theorem 4.10 respectively.2. Basic Set Up
This section entails basic definitions and concepts which make this article self-contained. Throughout this paper, the character k represents a field, and R is apolynomial ring over k in n variables x , x , . . . , x n . We start with the followingdefinition of a simplicial complex. Definition 2.1.
Let V = { v , v , ..., v n } be a vertex set and ∆ be a subset of P ( V ).We say ∆ a simplicial complex on V if, ( i ) { v i } ∈ ∆ for all i ∈ { , , . . . , n } , and,( ii ) subsets of every element of ∆ belong to ∆.The members of ∆ are known as faces; the dimension of a face F is one less thanthe cardinality of F . The maximal faces under inclusion are known as facets. It isclear that a simplicial complex can be determined by its facets. If F , F , . . . , F r arethe facets of ∆, we write ∆ = h F , F , ..., F r i to say that ∆ is generated by these F ′ i s . The dimension of a simplicial complex ∆ is defined as follows:dim(∆) = max { dim( F ) | F is facet in ∆ } Remark 2.2.
A simplicial complex whose facets can have at most dimension 1 isactually a simple graph, where the 1-dimensional facets are termed as edges, andthe 0-dimensional facets are isolated vertices. A graph is usually denoted by G with E ( G ) being the set of its edges.We need to recall few more concepts from the literature before giving the definitionof (quasi) f -ideals. Definition 2.3.
The f -vector of a d -dimensional simplicial complex ∆ is an element( f , f , ..., f d ) ∈ Z d +1 , where f i = |{ F ∈ ∆ : dim( F ) = i }| for all i ∈ { , , , ..., d } .The f -vector of ∆ is denoted by f (∆) . Definition 2.4. ( facet complex and non-face complex ) Consider a square-free monomial ideal I of R = k [ x , x , . . . , x n ] with G ( I ) = { m , m , ..., m r } as itsunique minimal monomial system of generators. Then F , F , ..., F r are the facetsof the facet complex of I on the vertices v , v , . . . , v n , where F i = { v j : x j divide m i } , where i ∈ { , , ..., r } . The facet complex of I is denoted by δ F ( I ). And,the non-face complex of I is a simplicial complex on V = { v , v , . . . v n } such thata subset { v i , v i , ..., v i k } of V is a face of this non-face complex if and only if thecorresponding monomial x i x i · · · x i k does not belong to I . The non-face complexof I is also known as the Stanley-Reisner complex of I , and we denote it by δ N ( I ). efinition 2.5. ( facet ideal and non-face ideal ) Consider a simplicial ∆ onthe vertex V = { v , v , . . . , v n } which is generated by the facets F , F , . . . , F r . Thefacet ideal of ∆ is square-free monomial ideal of R = k [ x , x , . . . , x n ] which isminimally generated by the monomials m , m , ..., m r such that m i = Q v j ∈ F i x j , where i ∈ { , , ..., r } . The facet ideal of ∆ is denoted by I F (∆). And, the non-face idealof ∆ is another square-free monomial ideal of R such that any monomial x i x i ...x i k is in the non-face ideal if and only if the corresponding subset { v i , v i , ..., v i k } of V does not belong to the complex ∆. The non-face ideal of ∆ is also known as itsStanley-Reisner ideal, and it is written as I N (∆). Definition 2.6.
A square-free monomial ideal I of the polynomial ring R is saidto be an f -ideal if and only if the f -vector of the facet complex of I coincides withthe f -vector of its Stanley-Reisner complex, i.e. f ( δ F ( I )) = f ( δ N ( I )). A simplicialcomplex is said to be an f -simplicial complex if its facet ideal is an f -ideal. A1-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., quasi f -ideal, in the separate section. However, before moving to the next sec-tion, we would rather recall the definition of perfect sets of R . These sets are usedin characterizing f -ideals as given in [12, Theorem 2.3].Let Sm ( R ) denote the set of all square-free monomials in R ; let Sm ( R ) d be theset of all square-free monomials of degree d in Sm ( R ). For a subset T ⊆ Sm ( R ),consider the upper shadow, ⊔ ( T ), and the lower shadow of T , ⊓ ( T ), as given below: ⊔ ( T ) = { gx i | g ∈ T, x i does not divide g, ≤ i ≤ n }⊓ ( T ) = { h | h = g/x i for some g ∈ T and some x i with x i | g } If, in particular, T sits in Sm ( R ) d , then ⊔ ( T ) ⊂ Sm ( R ) d +1 and ⊓ ( T ) ⊂ Sm ( R ) d − .The set T is then called upper perfect if ⊔ ( T ) = Sm ( R ) d +1 , and it is said to be lowerperfect if its lower shadow is full, i.e., ⊓ ( T ) = Sm ( R ) d − . The set T is called a perfectset if and only if it is lower perfect as well as upper perfect. In general, perfect setscan have different cardinalities; for example, every subset of Sm ( R ) d containinga perfect set is again a perfect set. The smallest number among the cardinalitiesof perfect sets of degree d is called the ( n, d ) th perfect number, and is denoted by N ( n, d ). By [12, Lemma 3.3], for a positive t and n ≥
4, we have the followingequations: N ( n,
2) = (cid:26) t − t, when n = 2 t ; t , when n = 2 t + 1.3. Quasi f -Ideals: Definition and Examples In this section, we have introduced the notion of quasi f -ideals, quasi f -graphsand quasi f -simplicial complexes. Some examples are also presented. efinition 3.1. Let ( a , a , . . . , a s ) ∈ Z s . A square-free monomial ideal I in thepolynomial ring R = k [ x , x , ..., x n ] is said to be a quasi f -ideal of type ( a , a , . . . , a s )if and only if f ( δ N ( I )) − f ( δ F ( I )) = ( a , a , . . . , a s ).If I is quasi f -ideal of type ( a , a , . . . , a s ) in the ring R = k [ x , x , ..., x n ], thenthe definition requires that both the complexes of I , the facet complex and theStanley-Reisner complex, should be s -dimensional. It means that s ≤ n ; in fact, s + 1 = max { deg( u ) : u ∈ G ( I ) } , 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 -idealis a quasi f -ideal whose type is a zero vector, and obviously any quasi f -ideal withtype some non-zero vector can not be f -ideal. But we would like to mention thatthe 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 = k [ x , x , ..., x n ], forany n .Let us now consider some examples of quasi f -ideals of some types below. Example 3.2.
Every f -ideal is quasi f -ideal of type . We would like the readersto see [1],[2], [11],[12] and [13] to know more about f -ideals and f -graphs. Example 3.3.
The ideal J = h x x x , x x x , x x x , x x x , x x x i of the poly-nomial ring in 5 variables, which has been discussed in the introduction of this paper,is quasi f -ideal of type (0 , , Example 3.4.
Consider the monomial ideal I = h x x x , x x x , x x x , x x x ,x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x ,x x x , x x x i of degree 3 in the ring R = k [ x , x , x , x , x , x , x ]. The facet com-plex and the non-face complex of I are δ F ( I ) = h{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }i and δ N ( I ) = h{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }{ v , v , v } , { v , v , v } , { v , v , v } , { v , v , v } , { v , v , v }i this implies that f ( δ F ( I )) = (7 , ,
17) and f ( δ N ( I ))) = (7 , , I is a quasi f -ideal of the type (0 , , Quasi f -Ideals of Degree 2 Now we want to characterize those quasi f -ideals in the ring R = k [ x , x , ..., x n ]whose minimal generating set is a subset of the set of all square-free monomial ofdegree 2 in R . We may call such ideals as equigenerated (or pure) square-free mono-mial ideals of degree 2. Obviously, the type of such quasi f -ideals will be ordered pair( a, b ) ∈ Z . However, since we are to consider only those ideals for which the facet omplex and the non-face complex have the same vertex set V = { v , v , v , . . . , v n } ,so we are bound to consider only those ideals whose support is full, i.e. an ideal I of R = k [ x , x , ..., x n ] for which S u ∈ G ( I ) supp ( u ) = { x , x , . . . , x n } . This means thatif 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 theoremcharacterizes all such ideals. Theorem 4.1.
Let I be an equigenerated square-free monomial ideal of R = k [ x , x , ..., x n ] of degree 2, and let G ( I ) = { u , u , . . . , u r } be the unique mini-mal generating set of I . Then I is quasi f -ideal of type (0 , b ) ∈ Z if and only if thefollowing conditions hold true:(1) (i) ht ( I ) = n − (cid:0) n (cid:1) ≡ ( mod
2) if b is even1 ( mod
2) if b is odd ;(3) | G ( I ) | = ( (cid:0) n (cid:1) − b ). Proof.
First suppose that I is quasi f -ideal of type (0 , b ). This means that dim ( δ F ( I ))= dim ( δ N ( I )) and f ( δ N ( I )) − f ( δ F ( I )) = (0 , b ). But as I is equigenerated square-free monomial ideal of degree 2, we have that dim ( δ F ( I )) = 1. Also, from [16,Corollary 6.3.5], we know that dim ( δ N ( I ))) = n − ht ( I ) −
1. Now the equality of di-mensions of these two complexes yields that ht ( I ) = n −
2. Moreover, as | G ( I ) | = r , f ( δ F ( I )) = r . Therefore, by [1, Lemma 3.2], we have f ( δ N ( I )) = (cid:0) n (cid:1) − r .The fact that I is a quasi f -ideal of the type (0 , b ) also gives us the equation: f ( δ N ( I )) − f ( δ F ( I )) = b , hence, (cid:0) n (cid:1) − r − r = b or (cid:0) n (cid:1) − r = b . This furtherimplies that | G ( I ) | = r = ( (cid:0) n (cid:1) − b ). Note that the equation (cid:0) n (cid:1) − r = b also tellsthat the parity of (cid:0) n (cid:1) is same as the parity of b , and hence the condition ( ii ).Conversely, we suppose that the conditions ( i ), ( ii ) and ( iii ) are satisfied. As I is equigenerated square-free monomial ideal of degree 2, dim ( δ F ( I ))) = 1. Thecondition ( i ) together with the fact that dim ( δ N ( I ))) = n − ht ( I ) − δ N ( I ) also has dimension 1. As both the complexes, δ F ( I ) and δ N ( I ),have the same vertex set V = { v , v , . . . , v n } , so f ( δ F ( I ) = f ( δ N ( I ) = n . Thisshows that f ( δ N ( I )) − ( f ( δ F ( I )) = 0. The other two conditions together with [1,Lemma 3.2] give the following: f ( δ N ( I )) − ( f ( δ F ( I )) = (cid:18) n (cid:19) − | G ( I ) | − | G ( I ) | = (cid:18) n (cid:19) − | G ( I ) | = (cid:18) n (cid:19) − (cid:18) n (cid:19) − b ) = b Thus I is quasi f -ideal of type (0 , b ). (cid:3) Remark 4.2.
While characterizing f -ideals of degree 2, we were bound to dealwith only those polynomial rings in n variables for which (cid:0) n (cid:1) was even. But thisrestriction is no longer needed for the case of quasi f -ideals of degree 2, as theabove theorem also considers the situation when (cid:0) n (cid:1) is odd. However, it imposessome restrictions on the value of b ; it says that if I is quasi f -ideal of degree 2 in he polynomial ring R = k [ x , x , ..., x n ] having type (0 , b ), then the parity of (cid:0) n (cid:1) and b must be same. So, if n = 4 k or n = 4 k + 1 and b is odd, then there willnot be any quasi f -ideal of degree 2 of type (0 , b ). Similarly for n = 4 k + 2 or n = 4 k + 3, there will not be any quasi f -ideal of type (0 , b ) with even b . Moreover,since b = f ( δ N ( I )) − f ( δ F ( I )), [1, Lemma 3.2] shows that | b | can not be greaterthan (cid:0) n (cid:1) . Also, as the height of a quasi f -ideal of degree 2 has to be n − | b | 6 = (cid:0) n (cid:1) .The next theorem gives a combinatorial characterization of quasi f -ideals of degree2. This theorem involves the notion of an upper perfect set. Theorem 4.3.
Let I be an equigenerated square-free monomial ideal of the poly-nomial ring R = k [ x , x , ..., x n ] of degree 2, and let G ( I ) be its minimal generatingset. Then I is quasi f -ideal of type (0 , b ) ∈ Z (where | b | < (cid:0) n (cid:1) ) if and only if thefollowing conditions are satisfied:(1) the parity of (cid:0) n (cid:1) is same as the parity of b ,(2) the set G ( I ) is upper perfect with | G ( I ) | = ( (cid:0) n (cid:1) − b ). Proof.
Let us first suppose that I is quasi f -ideal of type (0 , b ). The condition (1)is the condition (2) of Theorem 4.1; clearly, | G ( I ) | = ( (cid:0) n (cid:1) − b ), as I is a quasi f -ideal of type (0 , b ). Now we only have to show that G ( I ) is upper perfect. This isequivalent to say that I contains all square-free monomial of degree 3. Indeed it is so,otherwise if there is some monomial (say) x i x i x i which does not belong to I , thenthe corresponding subset { v i , v i , v i } ∈ δ N ( I ). This means that dim ( δ N ( I )) ≥ dim ( δ N ( I )) = 1.Conversely, note that the conditions (2) and (3) of Theorem 4.1 directly followsfrom the assumption. We only have to show that ht ( I ) = n −
2. As the ideal I contains all square-free monomials of degree 3 and higher, so dim ( δ N ( I )) ≤
1. Thefact that | b | < (cid:0) n (cid:1) yields that δ N ( I )) is 1-dimensional complex, which implies that ht ( I ) = n −
2. Thus I is quasi f -ideal of type (0 , b ). (cid:3) It will be interesting to determine the bounds on the values of b , which is givenin the proposition below. Proposition 4.4.
Let I be a quasi f -ideal of degree 2 and type (0 , b ) in the poly-nomial ring R = k [ x , x , ..., x n ]. Then the following holds true: − (cid:18) n (cid:19) + 2 ≤ b ≤ (cid:18) n (cid:19) − N ( n, Proof. As I is quasi f -ideal of degree 2 and type (0 , b ), the theorem 4.3 tells us that G ( I ) is a perfect subset of Sm ( R ) with | G ( I ) | = ( (cid:0) n (cid:1) − b ). Since N ( n,
2) is thesmallest cardinality of perfect sets of degree 2, this means that | G ( I ) | ≥ N ( n, ⇒
12 ( (cid:18) n (cid:19) − b ) ≥ N ( n, ⇒ b ≤ (cid:18) n (cid:19) − N ( n, oreover, it is clear that | G ( I ) | ≤ (cid:0) n (cid:1) . This means that ( (cid:0) n (cid:1) − b ) ≤ (cid:0) n (cid:1) , whichgives the inequality: − (cid:0) n (cid:1) ≤ b . However, if b = − (cid:0) n (cid:1) , then | G ( I ) | = (cid:0) n (cid:1) . Thesquare-free monomial ideal of degree 2 with (cid:0) n (cid:1) generators has height n −
1. Sincethe quasi f -ideal of degree 2 and type (0 , b ) should be of height n −
2, so the value b = − (cid:0) n (cid:1) is not acceptable. Also, as the parity of b and − (cid:0) n (cid:1) has to be same, theimmediate acceptable value of b which is greater than − (cid:0) n (cid:1) would be − (cid:0) n (cid:1) + 2.The value b = − (cid:0) n (cid:1) + 2 can be realized for by any square-free monomial ideal ofdegree 2 whose generating set consists of (cid:0) n (cid:1) − − (cid:0) n (cid:1) + 2 ≤ b ≤ (cid:0) n (cid:1) − N ( n, (cid:3) Now let us talk about the associated prime ideals of these ideals. Consider a quasi f -ideal I of degree 2 and type (0 , b ) and let p be any minimal prime ideal of I . Bythe condition ( i ) of Theorem 4.1, we have that ht ( p ) ∈ { n − , n − } . In the nexttheorem, we see that which monomial prime ideals of height n − n − Ass ( R/I ). Theorem 4.5.
Let I be a quasi f -ideal of degree 2 and type (0 , b ). Then thefollowing statements are true:( i ) A monomial prime ideal p of height n − Ass ( R/I ) if and only if thesquare-free quadratic monomial x i x j / ∈ G ( I ), where x i , x j / ∈ p .( ii ) A monomial prime ideal p of height n − Ass ( R/I ) if and only if x i x j ∈ G ( I ) for all j , where x i / ∈ p . Proof.
We will use the well-known one-to-one correspondence between facets of δ N ( I ) and the minimal vertex covers of δ F ( I ), which correspond to the minimalprimes of I (as given in [6]). More precisely, F is a facet of δ N ( I ) if and only if p = ( x i | i / ∈ F ) is a minimal prime of I .Case ( i ): Since I is square-free monomial ideal, the associated primes of I are pre-cisely the minimal primes. Thus, a monomial prime ideal p of height n − Ass ( R/I ) if and only if { v i , v j } is a facet of δ N ( I ), where x i , x j / ∈ p . This isequivalent to say that x i x j / ∈ G ( I ), because I is quasi f -ideal of type (0 , b ) anddegree 2.Case ( ii ): Let p be a monomial prime ideal of height n − x i / ∈ p . Then p belongs to Ass ( R/I ) if and only if { v i } is a facet (isolated vertex) of δ N ( I ). Thisis equivalent to say that the sets { v i , v j } do not belong to δ N ( I ) for all j = i .Equivalently, x i x j ∈ G ( I ) for all j = i , because G ( I ) ⊂ Sm ( R ) . (cid:3) We now move ahead and describe how quasi f -ideals of degree 2 can be con-structed. In particular, we show that every ordered pair (0 , b ), where − (cid:0) n (cid:1) + 2 ≤ b ≤ (cid:0) n (cid:1) − N ( n, f -ideal of degree 2. Proposition 4.6.
Let A be a proper subset of { x , x , . . . , x n } . Then for the subset P A = { x i x j | x i , x j ∈ A or x i , x j / ∈ A } of Sm ( R ) , the ideal I = h P A i is a quasi f -ideal in R = k [ x , x , . . . , x n ] of degree and type (0 , n − ( n − | A | ) ) . roof. It is easy to see that G ( I ) = W A is a perfect set of Sm ( R ) . Moreover, as | W A | = (cid:0) | A | (cid:1) + (cid:0) n −| A | (cid:1) = | A | − n | A |− n + n = ( (cid:0) n (cid:1) − n − ( n − | A | ) ), the theorem 4 . I is a quasi f -ideal of the type (0 , n − ( n − | A | ) ). (cid:3) Proposition 4.7.
Let b be an integer satisfying the inequality of Proposition 4.4.Then the pair (0 , b ) can be realized as the type of some quasi f -ideal provided b and (cid:0) n (cid:1) have same parity. Proof. If (cid:0) n (cid:1) is even, then either n = 4 t or n = 4 t + 1. Similarly, if (cid:0) n (cid:1) is odd, theneither n = 4 t + 2 or n = 4 t + 3. We consider each situation case by case.Case ( i ): n = 4 t . For the subset A = { x , x , . . . , x t } , we claim that the ideal I = < W A ∪ D > is quasi f -ideal of type (0 , b ), where D is any subset of Sm ( R ) − W A such that | D | = ( (cid:0) n (cid:1) − b ) − | W A | . It is enough to show that ( (cid:0) n (cid:1) − b ) ≥ | W A | ,because only then we can form such D . Note that | W A | = 2 (cid:0) t (cid:1) = 4 t − t . Theexpression ( (cid:0) n (cid:1) − b ) assumes the least value when b takes the largest value whichis b = (cid:0) n (cid:1) − N ( n, n = 4 t is even, the lemma 3.3 of [12] tells that b = (cid:0) t (cid:1) − t − t ) = 2 t . This means that ( (cid:0) n (cid:1) − b ) = ( (cid:0) t (cid:1) − t ) = 4 t − t = | W A | .Consequently, if b < (cid:0) n (cid:1) − N ( n, ( (cid:0) n (cid:1) − b ) > | W A | . Case ( ii ): n = 4 t + 1. Consider the same subset A = { x , x , . . . , x t } and thesame ideal I = < W A ∪ D > , where D is any subset of Sm ( R ) − W A such that | D | = ( (cid:0) n (cid:1) − b ) − | W A | . In this case, | W A | = (cid:0) t (cid:1) + (cid:0) t +12 (cid:1) = 4 t . Once again theleast value of the expression (cid:0) n (cid:1) − N ( n,
2) is assumed for the largest b = (cid:0) n (cid:1) − N ( n, n = 4 t + 1 is odd, we have b = (cid:0) n (cid:1) − N ( n,
2) = (cid:0) t +12 (cid:1) − t ) =2 t . This means that ( (cid:0) n (cid:1) − b ) = ( (cid:0) t +12 (cid:1) − t ) = 4 t = | W A | . Finally, we have ( (cid:0) n (cid:1) − b ) > | W A | for each b which is strictly less then (cid:0) n (cid:1) − N ( n, A = { x , x , . . . , x t +1 } , it can be shown that ( (cid:0) n (cid:1) − b ) ≥ | W A | for the cases n = 4 t + 2 and n = 4 t + 3. Now choosing any subset D of Sm ( R ) − W A such that | D | = ( (cid:0) n (cid:1) − b ) − | W A | and setting G ( I ) = W A ∪ D gives us quasi f -ideal I of type(0 , b ). (cid:3) Example 4.8.
For n = 8, we have − ≤ b ≤
4. In order to construct quasi f -ideal I of type (say) (0 , − A = { x , x , x , x } . Then W A = { x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x } . Note that | W A | = 12. Also, as | G ( I ) | shouldbe equal to ( (cid:0) (cid:1) − ( − D of Sm ( R ) − W A consisting of five elements. Let it be D = { x x , x x , x x , x x , x x } . Then I = h W A ∪ D i is a quasi f -ideal of degree 2 and type (0 , − G ( I ) is perfectand | G ( I ) | = 17 = ( (cid:0) (cid:1) − ( − . Note that for any even b satisfying − ≤ b ≤ f -ideal of type (0,b) following the same procedure. Theorem 4.9.
Let I be a quasi f -ideal in R = k [ x , x , ..., x n ] of the type (0 , b ) . Then the Hilbert function of
R/I is given by H ( S/I, z ) = 14 { ( n − n + 2 b ) z − n + 5 n − b } Proof.
Suppose that I is a quasi f -ideal of the type (0 , b ), hence, f ( δ N ( I )) = | G ( I ) | + b . Therefore, f ( δ N ( I )) = ( n, | G ( I ) | + b ). By [16, Proposition 6.7.3], the Hilbert unction of R/I is given by H ( K [∆] , z ) = d X i =0 (cid:18) z − i (cid:19) f i = (cid:18) z − (cid:19) f + (cid:18) z − (cid:19) f = n + ( z − | G ( I ) | + b )On simplification, we get H ( S/I, z ) = n + ( z − n ( n − b ) = 14 { ( n − n + 2 b ) z − n + 5 n − b } (cid:3) Theorem 4.10.
Let I be a quasi f -ideal in R = k [ x , x , ..., x n ] of the type (0 , b ) . Then the Hilbert series of
R/I is given by F ( R/I, z ) = 4 + 4( n − z + ( n − n + 4 + 2 b ) z − z ) Proof.
Suppose that I is a quasi f -ideal of the type (0 , b ), the definition impliesthat f ( δ N ( I )) − f ( δ F ( I )) = b , which further means that f ( δ N ( I )) = | G ( I ) | + b .Therefore, f ( δ N ( I )) = ( n, | G ( I ) | + b ). Now using [16, Theorem 6.7.2], we have that F ( R/I, z ) = f − z (1 − z ) + f z (1 − z ) + f z (1 − z ) = 1 + nz (1 − z ) + ( | G ( I ) | + b ) z (1 − z ) and after simplification, we get F ( R/I, z ) = 1 + ( n − z + (1 − n + | G ( I ) | + b ) z (1 − z ) which implies the desired formula. (cid:3) References [1] Abbasi, G.Q., Ahmad, S., Anwar, I. and Baig, W.A., 2012. f-Ideals of degree 2. Algebra Collo-quium, 19 (01) 921-926.[2] Anwar, I., Mahmood, H., Binyamin, M.A. and Zafar, M.K., 2014. On the Characterization off-Ideals. Communications in Algebra, 42(9) 3736-3741.[3] Bruns, W., Herzog, J., 1998. Cohen-Macaulay Rings, Cambridge University Press.[4] Buchstaber, V.M., Panov, T.E., 2015. Toric Topology, American Mathematical Society.[5] Budd, S., Van Tuyl, A., 2019. Newton Complementary Dual of f -ideals. Canad. Math. Bull.62 (2) 231-241.[6] Faridi, S., 2002. The Facet Ideal of a Simplicial Complex. Manuscripta Mathematica, 109(2)159-174.[7] Francisco, C. A., Mermin, J., Schweig, J., 2014. A survey of Stanley-Reisner theory. ConnectionsBetween Algebra, Combinatorics, and Geometry. Springer Proc. Math. Stat., Springer, NewYork, 76 209-234.[8] Herzog, J. and Hibi, T., 2018. Binomial Ideals, Springer London.[9] Herzog, J. and Hibi, T., 2011. Monomial ideals, Springer London.[10] Guo, J., Wu, T. and Liu, Q., 2013. Perfect Sets and f -Ideals. arXiv preprint arXiv:1312.0324.[11] Guo, J., Wu, T.,2015. On the ( n, d ) th f -ideals, J. Korean Math. Soc. 52(4), pp.685-697.
12] Guo, J., Wu, T. and Liu, Q., 2017. f -Ideals and f -Graphs. Communications in Algebra, 45(8)3207-3220.[13] Mahmood, H., Anwar, I. and Zafar, M.K., 2014. A Construction of Cohen-Macaulay f-Graphs.Journal of Algebra and Its Applications, 13(06), 1450012-1450019.[14] Mahmood, H., Anwar, I., Binyamin, M.A. and Yasmeen, S., 2016. On the Connectedness off-Simplicial Complexes. Journal of Algebra and Its Applications, p.1750017.[15] Mahmood, H., Rehman, F.U., Binyamin, M.A., 2019. A Note on f -Graphs (to appear inJournal of Algebra and Its Applications).[16] Villarreal, R., 2015. Monomial algebras, CRC Press.-Graphs (to appear inJournal of Algebra and Its Applications).[16] Villarreal, R., 2015. Monomial algebras, CRC Press.