# Join of two elements

Let be two elements with atoms. The pair satisfies the following universal property: supports both and , and if is another element with this property, then supports .

We demonstrate the existence of an element denoted , with a dual property: is supported both by and by , and if is another element with this property, then is supported by .  The construction does not require any assumptions on the symmetry.

Recall that an element supports an element if for any permutation of the atoms, implies . Equivalently, there is an equivariant mapping which maps to . If supports and supports , then and are equivariantly equivalent. The element satisfying the  above universal property is unique, up to equivariant equivalence.

Example. If are distinct atoms, then is equivariantly equivalent to . On the other hand, is equivariantly equivalent to .

We now present the construction of in general. It is vaguely inspired by the construction of a pushout using coproducts and coequalizers.

Let and be two elements. Let denote the orbit of and denote the orbit of . There is a smallest equivariant relation on the disjoint union such that . Let denote the -equivalence class of the element .

Claim. is supported both by and by , and if is another element with this property, then is supported by .

Proof. Since is equivariant, it defines an equivariant mapping from to , which assigns to an element its equivalence class. In particular, is supported by . Similarly, is supported by .

Now, let be any element which is supported both by and by . Let be the orbit of . Then there is a unique equivariant mapping such that , and a unique equivariant mapping such that . Consider the equivalence relation on such that for iff .

Observe that there is an equivariant mapping from the set of -equivalence classes to , and that this mapping maps the -equivalence class of to .

This relation is equivariant and since , it follows that it is coarser than the relation . In particular, there is an equivariant mapping which maps the -equivalence class of an element to its -equivalence class.

It follows that there is an equivariant mapping which maps the element , i.e., the -equivalence class of , to .