Shrinking braids and Left distributive monoid

TL;DR

This paper introduces shrinking braids, defines a related monoid, and extends the Dehornoy order to this new structure, establishing an isomorphism with a geometrically generated monoid.

Contribution

It presents a novel generalization of braids called shrinking braids and constructs a new monoid with an extended order structure.

Findings

Defined the relations of shrinking braids.

Extended the Dehornoy order to the monoid R.

Proved the isomorphism between R and the geometrically generated monoid.

Abstract

We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a \emph{left distributive monoid} structure and use it to extend the Dehornoy order on $B_{\infty}$ to an order on $R$. By using this order, we prove that $R$ is isomorphic to the monoid which is generated (geometrically) by shrinking braids.