A Gr&amp;#246;bner-Shirshov basis over a special type of braid monoids

Ahmet S Cevik, ,Hamed H Alsulami,Eylem G Karpuz,Esra K Cetinalp,Esra K Cetinalp,Esra K Cetinalp

doi:10.3934/math.2020278

Abstract

The aim of this paper is to present a Grobner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid In, in terms of the dex-leg ordering on the related elements of monoid. By taking into account the Grobner-Shirshov basis, the ideal form (or, equivalently, the normal form structure) of this important monoid will be obtained. This ideal form will give us the solution of the word problem. At the final part of this paper, we give an application of our main result which find out a Grobner-Shirshov basis for the symmetric inverse monoid I4 such that the accuracy and efficiency of this example can be seen by GBNP package in GAP (Group, Algorithms and Programming) which computes Grobner bases of non-commutative polynomials [ 1 ].

Highlights

Introduction and preliminariesThe Gröbner basis theory for commutative algebras was introduced by Buchberger [2] which provided a solution to the reduction problem for commutative algebras
We may refer the papers [6,7,8,9,10,11,12,13] for some recent studies over Gröbner-Shirshov bases in terms of algebraic ways, the papers [14, 15] related to Hilbert series and the paper [16] in terms of graph theoretic way
Our aim in this paper is to find a Gröbner-Shirshov basis of the symmetric inverse monoid in terms of the dex-leg ordering on the related words of symmetric inverse monoids

Summary

Introduction and preliminaries

The Gröbner basis theory for commutative algebras was introduced by Buchberger [2] which provided a solution to the reduction problem for commutative algebras. The method Gröbner-Shirshov basis theory gives a new algorithm to obtain normal forms of elements of groups, monoids and semigroups, and a new algorithm to solve the word problem in these algebraic structures (see [19], for relationship with word problem for semigroups and ideal membership problem for non-commutative polynomail rings). By considering this fact, our aim in this paper is to find a Gröbner-Shirshov basis of the symmetric inverse monoid in terms of the dex-leg ordering on the related words of symmetric inverse monoids. In [23], the author has studied presentations of symmetric inverse and singular part of the symmetric inverse monoids

Gröbner-Shirshov bases and the composition-diamond lemma

A Gröbner-Shirshov basis for the monoid In

An application

Conclusions