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For a family of height one orders $(X,le)$ and each non-degenerate solution $r_0colon Xtimes Xlongrightarrow Xtimes X$ of the set-theoretic braid equation on $X$ satisfying suitable conditions, we obtain all the non-degenerate solutions of the braid equation on the incidence coalgebra of $(X,le)$ that extend $r_0$.
In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-bra
In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in gen
We study Artin-Tits braid groups $mathbb{B}_W$ of type ADE via the action of $mathbb{B}_W$ on the homotopy category $mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following
To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permut
Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $nin N$. We say that such representations ar