Hey I have a terminology question. There are two ways to turn a preorder into a equivalence relation. 1) the symmetric closure:

\\[a \sim b := a \le b \text{ or } b \le a \\]

and 2) the symmetric ??:

\\[a \sim b := a \le b \text{ and } b \le a \\]

I think I've read "core" used for this in the context of turning groups commutative. Is that a good choice for the general idea? The closure is often the free construction, and I think the "core" would be the other adjoint to the forgetful functor some times.

\\[a \sim b := a \le b \text{ or } b \le a \\]

and 2) the symmetric ??:

\\[a \sim b := a \le b \text{ and } b \le a \\]

I think I've read "core" used for this in the context of turning groups commutative. Is that a good choice for the general idea? The closure is often the free construction, and I think the "core" would be the other adjoint to the forgetful functor some times.