The monotone, linear, etc. functions are called clones and the clones form a lattice called Post's Lattice https://en.wikipedia.org/wiki/Post%27s_lattice https

The monotone, linear, etc. functions are called clones and the clones form a lattice called Post's Lattice https://en.wikipedia.org/wiki/Post%27s_lattice https://en.wikipedia.org/wiki/Post%27s_lattice Given several logic gates, if their most recent common ancestor in the lattice is the clone of all boolean functions, then that set is universal. This provides a convenient prescription for deciding universality for a new k-input gate.