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

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


Introduction
Hilbert series of a graded commutative algebra is strongly related with dimensions of the homogeneous components of the algebra. is 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.
is 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 X � X 1 , X 2 , . . . , X n and words W � W 1 , W 2 , . . . , W d , then it is known that the numbers of words on X avoiding this word W is in one-to-one correspondence with monomials in X having nonzero image in the ring L � C[X]/W. If L 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]. ings become interesting especially when the monoid L has a rational Hilbert series. is happens exactly when we have a finitely presented monoid. Significance of Hilbert series can be described in terms of the growth of a monoid. e growth of a monoid is said to be polynomially bounded if the nth coefficient of its Hilbert series is 0(n d ) for some number d. A monoid has exponential growth if the nth coefficient is larger than C n for some C > 1. Interestingly, monoids should have polynomially bounded growths. e 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 n-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. e 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. ese 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. ese polytopes are convex-hull of some points in R n . Coxeter groups have generators a i , i ∈ I and have relations a 2 i � 1 and a i a j a i � a j a i a j with i, j ∈ I; 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 a 2 i � 1 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].
at is why, they are known as spherical. e Artin braid group, A n , is a spherical Coxeter groups. e 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 MB n which lead to find Hilbert series of MB n . In [7], we computed Hilbert series of MB 4 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 M(A ∞ n ) and showed that its growth rate is unbounded. In [12], we found the Hilbert series of M(D ∞ n ) and showed that its growth rate is also unbounded.
In this paper, we study the star topology S n and find recurrence relations and the Hilbert series of the associated right-angled monoid M(S ∞ n ). We also compute growth rate of the monoid M(S ∞ n ) and observe that it is unbounded.

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. ese basic preliminary facts and notations which will be required later for formulating our main results.
If the Coxeter group is finite, then A is called a spherical Artin group.

Definition 5.
e 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 ∞.
Definition 6 (see [14]). e length of a word g � s 1 , · · · , s n of a finitely generated group G is the smallest nonnegative integer n for which s 1 , . . . , s n ∈ S ∪ S − 1 , where S is the set of generators of G.
Definition 7 (see [14]). e spherical growth series of a finitely generated group G is H G (t) � ∞ k�0 a k t k , where a k is the number of words of length k.
Let a � b be a relation in a given monoid M. en, in length-lexicographic order, a is greater then or equal to b. A word uwv has an ambiguity if uw and wv are left sides two relations. If α 1 v and uα 2 are identical, then uwv is solvable. If α 1 v and uα 2 differ by lexicographic order, then we get a new relation in M. 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 w does not contain the LHS of any relation, then w is called a canonical word. e following notions are in [15][16][17][18][19][20] under different terminologies: Gröbner bases, complete presentation, rewriting system, and so on.

Main Results
In this part, we compute our main results.

Recurrence Relation of the Monoid M(S ∞ n ).
In this paper, we study the star topology S n and find recurrence relations and the Hilbert series of the associated right-angled monoid M(S ∞ n ). We compute the growth rate of the monoid M(S ∞ n ), and using the graph, we show that it is unbounded. e Coxeter graph of the star topology S n is given by the following graph (Figure 2): We denote the right-angled monoid associated with S n by M(S ∞ n ). In M, we fix a total order x 1 < x 2 < · · · < x n on the generators. Hence, clearly we have the following lemma.

Lemma 1.
e monoid M(S ∞ n ) has generators x 1 , x 2 , . . . , x n and relations is section covers some useful results about recursive relations of M(S ∞ n ): Consider a system [21] of linear relations: u i (t + 1) � a i1 (t)u 1 (t) + a i2 (t)u 2 (t) + · · · + a in (t)u n (t) (2) e solution of the system where λ i and u i , 1 ≤ i ≤ k are, respectively, the eigenvalues and eigenvectors of A(t).
e largest eigenvalue represents the growth rate of the sequence u 1 (t), u 2 (t), . . . , u k (t).
In the following, by c k and c k;i , we shall mean the number of canonical words of length k and words starting with x i .
Let S n (λ) denote the characteristic polynomial, then we have the following.

Theorem 1.
e characteristic polynomial S n (λ) of the system of recursive relations of M(S ∞ n ) satisfies the relation: where n ≥ 2 and S 1 (λ) � λ − 1.
Proof. e 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 : · · · : 4 · · · : · · · (E n ) n=6,7,8 : H 3 : x 5 x n x n x n x n-2 x n-1 x n x n-1 x 4 (I 2 (p)) p≥5, p≠6 : Figure 1: Spherical Coxeter graphs.
x n x n-1 x 1 x 2 x 3 x 4 x 5 S n :

Lemma 3.
In M(S ∞ n ), characteristic polynomial is given explicitly by Proof. From equation   7, and r 120 � 34.2. We have the following graph representing the behavior of the growth rate of S n (λ) (Figure 3) We observe that the growth rate for M(S ∞ n ) 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 M(S ∞ n ) is unbounded.

Data Availability
No such data are used in this research.

Conflicts of Interest
e authors declare that they have no conflicts of interest.