Lattice of clones

Remarks

The intersection of clones is a clone, since each clause of the definition of a clone is kept under intersections, so $\mathcal{L}_A$ is a complete lattice. Example: on $\set{0, 1}$, the meet of the clones generated by $\wedge$ and by $\vee$ is the clone of projections. Their join is the clone of all monotone operations that preserve both $0$ and $1$. Check for the meet: the members of the first clone are the meets $x_{i_1} \wedge \cdots \wedge x_{i_k}$, and those of the second are joins. A meet of $k \ge 2$ distinct variables differs from every join on some input, so only projections are in both. Non-example: the join is not the union in general. On $\set{0, 1}$, the union of the clones generated by $\wedge$ and by $\vee$ is not a clone, since it misses the majority operation $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, which lies in their join. The lattice of clones on any set is an algebraic lattice whose compact elements are the finitely generated clones.