> Why cannot g be just

> $$g(x) = \frac{x}{10}$$

> That is, without the floor?

The problem is that \$$x/10\$$ is not usually a natural number when \$$x\$$ is a natural number. You're looking for the right adjoint of a function \$$f : \mathbb{N} \to \mathbb{N}\$$. This must be a function \$$g : \mathbb{N} \to \mathbb{N}\$$, that is, a function that takes natural numbers and gives natural numbers. So, \$$g(x) = \frac{x}{10} \$$ can't possibly work!

Remember the definition:

**Definition.** Given preorders \$$(A,\le_A)\$$ and \$$(B,\le_B)\$$, a **Galois connection** is a monotone map \$$f : A \to B\$$ together with a monotone map \$$g: B \to A\$$ such that

$$f(a) \le_B b \textrm{ if and only if } a \le_A g(b)$$

for all \$$a \in A, b \in B\$$. In this situation we call \$$f\$$ the **left adjoint** and \$$g\$$ the **right adjoint**.