Minimal clone

Remarks

A clone $C \ne J_A$ is minimal exactly when every non-projection operation in $C$ generates $C$. Example: on $\set{0, 1}$, the clone generated by $\wedge$ is minimal. Check: its $n$-ary members are the operations $x_{i_1} \wedge \cdots \wedge x_{i_r}$, and identifying variables in one with $r \ge 2$ distinct variables gives $x \wedge y$ back. Non-example: the clone generated by $\wedge$ and $\vee$ on $\set{0, 1}$ is not minimal. It properly contains the clone generated by $\wedge$, which is not the projection clone. On $\set{0, 1}$ there are exactly seven minimal clones, generated by the constant $0$, the constant $1$, $\neg$, $\wedge$, $\vee$, the majority operation, and $x + y + z$ modulo $2$.