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A Gröbner-Shirshov basis over a special type of braid monoids

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].