nLab operad of set operations

Let XX be a set. The usual operations intersection, union, symmetric difference, complement… on the subsets of XX can be presented in an homogeneous way using an operation Δi j n:𝒫(X) ×n𝒫(X)\multiscripts{_^n}{\Delta}{_i^j}:\mathcal{P}(X)^{\times n} \rightarrow \mathcal{P}(X) for every n1n \ge 1 and 0ijn0 \le i \le j \le n. The operad thus generated gives much more operations.

For every n1n \ge 1, A 1,...,A nA_{1},...,A_{n} subsets of XX, and 0ijn0 \le i \le j \le n, define the set: Δi j n(A 1,...,A n)={xA 1...A n,i|k[1,n],xA k|j} \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...A n=Δn n n(A 1,...,A n)A_{1} \cap ... \cap A_{n} = \multiscripts{_^n}{\Delta}{_n^n}(A_{1},...,A_{n})
  • A 1...A n=Δ1 n n(A 1,...,A n)A_{1} \cup ... \cup A_{n} = \multiscripts{_^n}{\Delta}{_1^n}(A_{1},...,A_{n})
  • A 1ΔA 2=Δ1 1 2(A 1,A 2)A_{1} \Delta A_{2} = \multiscripts{_^2}{\Delta}{_1^1}(A_{1}, A_{2})
  • =Δ1 1 2(A 1,A 1)\emptyset = \multiscripts{_^2}{\Delta}{_1^1}(A_{1},A_{1})
  • X=Δ0 n n(A 1,...,A n)X = \multiscripts{_^n}{\Delta}{_0^n}(A_{1},...,A_{n})
  • XA 1=Δ0 0 1(A 1)X - A_{1} = \multiscripts{_^1}{\Delta}{_0^0}(A_{1})
  • A 1=Δ1 1 1(A 1)A_{1} = \multiscripts{_^1}{\Delta}{_1^1}(A_{1})

We also have:

  • X(A 1...A n)=Δ0 0 n(A 1,...,A n)X - (A_{1} \cup ... \cup A_{n}) = \multiscripts{_^n}{\Delta}{_0^0}(A_{1},...,A_{n})
  • X(A 1...A n)=Δ0 n1 n(A 1,...,A n)X - (A_{1} \cap ... \cap A_{n}) = \multiscripts{_^n}{\Delta}{_0^{n-1}}(A_{1},...,A_{n})
  • Δ1 1 n(A 1,...,A n)\multiscripts{_^n}{\Delta}{_1^1}(A_{1},...,A_{n}) is the set of elements which are in exactly one A iA_{i}.

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

Created on August 3, 2022 at 21:48:02. See the history of this page for a list of all contributions to it.