More about quark Yukawa textures from selection rules without group actions

We study the coupling selection rules associated with non-group symmetries, i.e., ${\mathbb{Z}}_2$ gauging of ${\mathbb{Z}}_M$ symmetries. We clarify which Yukawa textures can be derived by our selection rules for $M=3,4$, and 5, and obtain various textures including the the nearest neighbor interaction type and its extension. Some of them cannot be realized by a conventional group-like symmetry. They lead to interesting phenomenology such as a solution to the strong CP problem without axion.