Operations that have adjoints rather than inverses

Question

Is there a name for the operation ⊖ which is the left adjoint of natural number addition in the following way?

c ⊖ d ≤ a ≡ c ≤ a + d

Is there a name for algebraic structures like this, whose operations have adjoints rather than inverses?

Answer 1

It seems an adjoint to the multiplication in a partially ordered monoid is called a Monus: https://en.wikipedia.org/wiki/Monus.

And this kind of structure in general is called a "semiring with monus", or an m-semiring.

Answer 2

This operation is just $\mathrm{max}(c-d,0).$ To get anything interesting happening, it's best not to be in a subcategory of a groupoid. The more general question is very broad, but for instance the theory of monoidal categories with duals covers many cases.