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In this paper, we prove that a subset of the non-commutative Gröbner basis for the braid group given by Bokut is a basis for the braid monoid in band generators. We give an inductive way of computing the infimum of a positive braid and hence a criteria to decide whether a positive word is prime to the fundamental braid or not. We also compute the Hilbert series of the braid monoid with three generators.

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids MBn+1. We also show that the Hilbert series of the positive words in MBn+1 with a given prefix are rational functions.

The concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

L. A. Bokut gave a Gröbner-Shirshov basis of the braid group Bn in band generators. Using this presentation and solving all the ambiguities we construct a linear system for irreducible words and compute the Hilbert series of the braid monoid M B4 .

In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.

ABSTRACT. In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given. .

We find the Hilbert series of the right-angled affine Artin monoid (̃∞). We also discuss its recurrence relation and the growth rate.