# Solutions of the braid equation and orders

## TL;DR

This paper explores how non-degenerate set-theoretic solutions of the braid equation on the underlying set of a locally finite order can be extended to solutions on its incidence coalgebra, advancing understanding of algebraic structures related to orders.

## Contribution

It introduces the concept of non-degenerate solutions on incidence coalgebras and initiates the study of their extension from set-theoretic solutions on orders.

## Key findings

- Defined non-degenerate solutions on incidence coalgebras
- Established conditions for extending set-theoretic solutions to coalgebras
- Laid groundwork for further research on algebraic structures of orders

## Abstract

We introduce the notion of a non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induce by restriction a non-degenerate set theoretic solution over the underlying set. So, it makes sense to ask if a non-degenerate set theoretic solution on the underlying set of a locally finite order extends to a non-degenerate solution on its incidence coalgebra. In this paper we begin the study of this problem.