Solutions of the Braid Equation and Orders

We introduce the notion of non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induces 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.