Journal of Mathematics
Volume 2020 (2020), Article ID 5215631, 6 pages
https://doi.org/10.1155/2020/5215631

Research Article

Recurrence Relations and Hilbert Series of the Monoid Associated with Star Topology

Jiang-Hua Tang,¹ Zaffar Iqbal,² Abdul Rauf Nizami,³ Mobeen Munir,⁴ Faiza Azam,² and Jia-Bao Liu⁵

¹Department of General Education, Anhui Xinhua University, Hefei 230088, China
²Department of Mathematics, University of Gujrat, Gujrat, Pakistan
³University of Central Punjab, Department of Mathematics, Lahore, Pakistan
⁴Department of Mathematics, Division of Science and Technology, University of Education Township, Lahore, Pakistan
⁵School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Correspondence should be addressed to Mobeen Munir; mmunir@ue.edu.pk

Received 1 April 2020; Accepted 21 July 2020; Published 27 August 2020

Academic Editor: Ghulam Shabbir

Copyright © 2020 Jiang-Hua Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Affine monoids are the considered as natural discrete analogues of the finitely generated cones. The interconnection between these two objects has been an active area of research since last decade. Star network is one of the most common in computer network topologies. In this work, we study star topology and associate a Coxeter structure of affine type on it. We find a recurrence relation and the Hilbert series of the associated right-angled monoid . We observe that the growth rate of the monoid is unbounded.

1. Introduction

Hilbert series of a graded commutative algebra is strongly related with dimensions of the homogeneous components of the algebra. This notion has been extended to filtered algebras and coherent sheaves over projective schemes [1]. Hilbert series is treated as a special case of Hilbert–Poincaré series of a graded vector space and is closely related with the number of words of an alphabet. Hilbert series helps counting words on an alphabet that do not contain a fixed set of words. This is also named as well-known forbidden subwords problem which can be translated at the level of monomial algebras. If we are given an alphabet and words , then it is known that the numbers of words on avoiding this word is in one-to-one correspondence with monomials in having nonzero image in the ring . If is a finitely generated monoid, then the coefficients of its Hilbert series satisfy a recursive relation. Hilbert series of several popular algebras appear as power series of rational functions [2]. Things become interesting especially when the monoid has a rational Hilbert series. This happens exactly when we have a finitely presented monoid. Significance of Hilbert series can be described in terms of the growth of a monoid. The growth of a monoid is said to be polynomially bounded if the th coefficient of its Hilbert series is for some number . A monoid has exponential growth if the th coefficient is larger than for some . Interestingly, monoids should have polynomially bounded growths. The smallest-degree polynomial bounding the growth is one less than the Krull dimension. In noncommutative settings, topologists used growth to study fundamental groups. Milnor proved that if a compact Riemannian manifold has all its sectional curvatures negative, then its fundamental group has exponential growth, and if a complete -dimensional Riemannian manifold has its mean curvature tensor everywhere positive semidefinite, then the finitely generated subgroup of its fundamental group has polynomially bounded growth. In short, a monoid has either an exponential growth or a polynomial growth. The degree of the growth is one less than the order of the pole at 1 of the Hibert series.

Coxeter groups were introduced by Canadian geometer H. S. M. Coxeter in 1934 to solve the well-known famous word problem, namely, whether two words occurring in generators of the presentation of groups correspond to same element or not. These groups have nice other properties such as having faithful linear representations as groups of reflections. In a nontrivial way, it can be proved that these groups are abstract analogues of the regular polytopes. These polytopes are convex-hull of some points in . Coxeter groups have generators and have relations and with ; finite and infinite groups are usually referred as spherical and affine.

Star topology is one of the important topologies used in networking and other real-world problems. One way to study this topology is by using Dynkin diagram [3] (or Coxeter graph), and other way is by using monoids. Note that the removal of the relations gives Artin groups. So, Coxeter groups are quotient groups of the Artin groups. A finite Coxeter group is a discrete acting group of reflections of a sphere [3]. That is why, they are known as spherical. The Artin braid group, , is a spherical Coxeter groups. The Infinite Coxeter groups are generated by reflections in affine spaces [3].

In 2009, Saito [4] found spherical growth series of Artin monoids [5]. In [6], we gave a linear system for the canonical words of the braid monoid which lead to find Hilbert series of . In [7], we computed Hilbert series of in band generators. In 2006, Mairesse and Mathéus [8] gave dihedral-type growth series of Artin groups. In 1993, Parry [9] gave the growth series of Coxeter groups. In [10], we proved that the upper bound of the growth of spherical Artin monoids is 4. But, in the affine case, this result is not true. In [11], we found a recurrence relation and Hilbert series of the associated right-angled affine Artin monoid and showed that its growth rate is unbounded. In [12], we found the Hilbert series of and showed that its growth rate is also unbounded.

In this paper, we study the star topology and find recurrence relations and the Hilbert series of the associated right-angled monoid . We also compute growth rate of the monoid and observe that it is unbounded.

2. Preliminaries

We start this section with the notion of Coxeter groups and Artin groups. We study the star topology as a Dynkin diagram and then convert it as a monoid. These basic preliminary facts and notations which will be required later for formulating our main results.

Definition 1. A square symmetric matrix is said to be a Coxeter matrix over a nonempty set such that all the diagonal entries are 1 and .

Definition 2. Let be a set of vertices of a labeled graph . We call a Coxeter graph if any two of its vertices are connected by an edge and if the label of each edge is greater than 2.
By convention, each edge is labeled only if the label is greater than 3.

Definition 3. A group with generator and relations and for all and is called a Coxeter group such that is the Coxeter matrix.

Definition 4. The Artin group is

If the Coxeter group is finite, then is called a spherical Artin group.

Definition 5. The right-angled Artin groups or monoids are obtained if all the labels, which are greater than or equal to 3, of spherical Coxeter graphs are replaced with .
The Artin spherical groups are usually represented by Coxeter graphs (see [3, 13]); these groups are for , for , for , , , , , , , , and for and . Figure 1 contains these graphs.

Figure 1: Spherical Coxeter graphs.

Definition 6. (see [14]). The length of a word of a finitely generated group is the smallest nonnegative integer for which , where is the set of generators of .

Definition 7. (see [14]). The spherical growth series of a finitely generated group is , where is the number of words of length .
Let be a relation in a given monoid . Then, in length-lexicographic order, is greater then or equal to . A word has an ambiguity if and are left sides two relations. If and are identical, then is solvable. If and differ by lexicographic order, then we get a new relation in . A presentation is said to be complete if solutions of all ambiguities are identical. A reducible word is the left side of a relation of a complete presentation of a monoid. If does not contain the LHS of any relation, then is called a canonical word. The following notions are in [15–20] under different terminologies: Gröbner bases, complete presentation, rewriting system, and so on.

3. Main Results

In this part, we compute our main results.

3.1. Recurrence Relation of the Monoid

In this paper, we study the star topology and find recurrence relations and the Hilbert series of the associated right-angled monoid . We compute the growth rate of the monoid , and using the graph, we show that it is unbounded. The Coxeter graph of the star topology is given by the following graph (Figure 2):

Figure 2: .

We denote the right-angled monoid associated with by . In , we fix a total order on the generators. Hence, clearly we have the following lemma.

Lemma 1. The monoid has generators and relations for and for .
This section covers some useful results about recursive relations of : Consider a system [21] of linear relations:

Equivalently, , where

The solution of the system is , where and are, respectively, the eigenvalues and eigenvectors of . The largest eigenvalue represents the growth rate of the sequence .
In the following, by and , we shall mean the number of canonical words of length and words starting with .

Lemma 2. satisfies the relations , , and    , where is

Let denote the characteristic polynomial, then we have the following.

Theorem 1. The characteristic polynomial of the system of recursive relations of satisfies the relation:

where and .

Proof. The characteristic polynomial of the coefficient matrix of the system of recurrence relations given in Lemma 2 is

Adding the last row in the 2nd last row, we have

Now, subtracting the 2nd last column from the last column and simplifying, we have , where

We write , where the determinants and are obtained by splitting such that the last row of is and the last row of is . Hence, easily, we have and . Therefore, we have

Here, we have an explicit formula for .

Lemma 3. In , characteristic polynomial is given explicitly by

Proof. From equation (5), we have . Hence, we have

3.2. Hilbert Series of the Monoid

From now on, will denote the Hilbert series of , and will denote the Hilbert series of of the words starting with .

Following Lemma 2 we get the following.

Theorem 2. For , we have

(1)(2)(3)

Proof.

(1)Since , .(2)Also, from Lemma 2, we have . Hence, .(3)It can be proved similarly.

The linear system of Theorem 2 takes the form , where

Lemma 4. For , we have .
For the system of equations given in Theorem 2, we have the following.

Lemma 5.

Proof. It is obvious to Just factor out from each row of .

Lemma 6.

Proof. The solution of the system is , where is the determinant of the matrix obtained by replacing 0 in the th column of by . That is,

Adding th column of to its column numbers , we get .
The following result gives the Hilbert series of .

Theorem 3.

Proof. Equation (5) gives . Hence, Lemma 6 and Theorem 2 imply that

3.3. Conclusions

Here, we gave, by constructing the affine-type Coxeter structure on star topology, a recursive relation and the Hilbert series of the right-angled monoid associated with a star graph. The main result is . We also computed the growth rate, , of using Mathematica; some initial values are: , , , , , , , , , , , , , , , , , . We also compute , , , , and . We have the following graph representing the behavior of the growth rate of (Figure 3)

Figure 3: .

We observe that the growth rate for increases and is unbounded. Hence, at the end, we have the following natural open problem emerging from our research.

An open problem: the growth rate of is unbounded.

Data Availability

No such data are used in this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported by the Natural Science Fund Project of Anhui Xinhua University ( Grant No. 2017zr011).

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