Vladislav wrote:

> 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**.

> 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**.