commutative operation in nLab

Contents

Idea

The notion of commutativity of an nn-ary operation in the sense that the order of the parameters can be permuted without changing the result. For n=2n = 2 this results in the notion of a commutative magma.

Definition

An n n -ary operation is a function f:A n→Af:A^n \to A from the cartesian power A nA^n to AA.

Let Fin(n)\mathrm{Fin}(n) denote the standard finite set with nn elements. Given a set AA, the Cartesian power A nA^n is isomorphic to the function set Fin(n)→A\mathrm{Fin}(n) \to A. As a result, one can equivalently define an nn-ary operation as a function f:(Fin(n)→A)→Af:(\mathrm{Fin}(n) \to A) \to A from the function set Fin(n)→A\mathrm{Fin}(n) \to A to AA. The function set Fin(n)→A\mathrm{Fin}(n) \to A represents the set of possible parameters of the nn-ary operation, and individual functions a:Fin(n)→Aa:\mathrm{Fin}(n) \to A represents individual possible parameters.

Using the function set definition, a permutation of the parameters of the nn-ary operation is the composition of the parameters a:Fin(n)→Aa:\mathrm{Fin}(n) \to A with a bijection h:Fin(n)≃Fin(n)h:\mathrm{Fin}(n) \simeq \mathrm{Fin}(n) on the standard finite set with nn elements.

Examples

Commuative binary operations

When n=2n = 2, the standard finite set with two elements Fin(2)\mathrm{Fin}(2) is in bijection with the boolean domain, which means we can simply use the boolean domain bool\mathrm{bool} in place of Fin(2)\mathrm{Fin}(2). There are two bijections on the boolean domain bool\mathrm{bool}, the identity function and the swap function which swaps the two elements around. By definition of the identity function, it is always true that f(a)=f(a∘id bool)f(a) = f(a \circ \mathrm{id}_{\mathrm{bool}}). Thus, the only relevant thing left to check is the swap function:

After applying the isomorphism between bool→A\mathrm{bool} \to A and the cartesian square A 2A^2 constructed from the recursion principle of the boolean domain, this definition results in the usual notion of commutative magma:

Commuative unary and nullary operations

By the above definition, every unary operation (n=1n = 1) is commutative because there is only one bijection on the standard finite set with one element, the identity function on Fin(1)\mathrm{Fin}(1), and f(a)=f(a∘id Fin(1))f(a) = f(a \circ \mathrm{id}_{\mathrm{Fin}(1)}). Similarly, every constant, considered as a nullary operation (n=0n = 0), is commutative because there is only one bijection on the standard finite set with zero elements, the identity function on Fin(0)\mathrm{Fin}(0), and for the unique function u:Fin(0)→Au:\mathrm{Fin}(0) \to A (since Fin(0)\mathrm{Fin}(0) is the initial set), f(u)=f(u∘id Fin(0))f(u) = f(u \circ \mathrm{id}_{\mathrm{Fin}(0)}).

Infinitary operations

The notion of a commutative operation can be extended from finitary operations to infinitary operations. The idea here is that an infinitary operation on a set AA consists of a set II of arities and a function f:(I→A)→Af:(I \to A) \to A. The notion of commutativity in the sense of permuting the parameters resulting in an equal element still makes sense for infinitary operations:

Examples of commutative infinitary operations include the II-indexed joins of a complete lattice for arbitrary set II, and the ℕ\mathbb{N}-indexed joins of a sigma-complete lattice.

• commutative magma