nLab operad of set operations

Let $X$ be a set. The usual operations intersection, union, symmetric difference, complement… on the subsets of $X$ can be presented in an homogeneous way using an operation $\multiscripts{_^n}{\Delta}{_i^j}:\mathcal{P}(X)^{\times n} \rightarrow \mathcal{P}(X)$ for every $n \ge 1$ and $0 \le i \le j \le n$. The operad thus generated gives much more operations.

For every $n \ge 1$, $A_{1},...,A_{n}$ subsets of $X$, and $0 \le i \le j \le n$, define the set: $\multiscripts{_^n}{\Delta}{_i^j}(A_{1},...,A_{n}) = \{x \in A_{1} \cup ... \cup A_{n}, i \le |k \in [1,n], x \in A_{k}| \le j\}$

We then have:

• $A_{1} \cap ... \cap A_{n} = \multiscripts{_^n}{\Delta}{_n^n}(A_{1},...,A_{n})$
• $A_{1} \cup ... \cup A_{n} = \multiscripts{_^n}{\Delta}{_1^n}(A_{1},...,A_{n})$
• $A_{1} \Delta A_{2} = \multiscripts{_^2}{\Delta}{_1^1}(A_{1}, A_{2})$
• $\emptyset = \multiscripts{_^2}{\Delta}{_1^1}(A_{1},A_{1})$
• $X = \multiscripts{_^n}{\Delta}{_0^n}(A_{1},...,A_{n})$
• $X - A_{1} = \multiscripts{_^1}{\Delta}{_0^0}(A_{1})$
• $A_{1} = \multiscripts{_^1}{\Delta}{_1^1}(A_{1})$

We also have:

• $X - (A_{1} \cup ... \cup A_{n}) = \multiscripts{_^n}{\Delta}{_0^0}(A_{1},...,A_{n})$
• $X - (A_{1} \cap ... \cap A_{n}) = \multiscripts{_^n}{\Delta}{_0^{n-1}}(A_{1},...,A_{n})$
• $\multiscripts{_^n}{\Delta}{_1^1}(A_{1},...,A_{n})$ is the set of elements which are in exactly one $A_{i}$.

We get an operad by starting with the operations $\multiscripts{_^n}{\Delta}{_i^j}$ and composing them together.