Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-Electric Condition

A local (or modified) Yang–Baxter equation (LYBE) gives a functional map from the parameters of the weights in the left-hand side to the parameters of the corresponding weights in the right-hand side of the LYBE. Such maps solve the functional tetrahedron equation. In this Letter, all the maps associated with LYBE's of the ferro-electric type with a single parameter in each weight matrix are classified.