A characterization of crossed self-similarity on crossed modules in L-algebras

We introduce crossed modules in cycloids, as a generalization of cycloids, which are algebraic logical structures arising in the context of the quantum Yang–Baxter equation. As a spacial case, we in particular focus on the crossed modules of $L-$algebras. These types of crossed modules are exceptional, since the category of $L-$algebras is not protomodular, nor Barr-exact, but it nevertheless has natural semidirect products that have not been described in category theoretic terms. We identify crossed ideals of crossed module in $L-$algebras, and obtain some characteristics of these objects that are normally not encountered on crossed modules of groups or algebras. As a consequence, we characterize crossed self-similarity completely in terms of properties of $L-$algebras and the boundary map forming the crossed module.