Galois connection

Remarks

Let $i \colon \Z \to \R$ be the inclusion and $\lfloor \cdot \rfloor \colon \R \to \Z$ the floor function. Then $i(n) \le x$ if and only if $n \le \lfloor x \rfloor$, so $(i, \lfloor \cdot \rfloor)$ is a Galois connection. On $\R$, the pair $f(x) = x + 1$, $g(y) = y$ is not a Galois connection: for $x = 0$ and $y = 1/2$, $x \le g(y)$ holds but $f(x) \le y$ fails. In a Galois connection both maps are monotone, $x \le g(f(x))$ and $f(g(y)) \le y$, and $f \circ g \circ f = f$ and $g \circ f \circ g = g$. Each adjoint determines the other: $g(y)$ is the greatest $x$ with $f(x) \le y$. The lower adjoint preserves all existing suprema, and the upper adjoint preserves all existing infima. An antitone Galois connection uses order-reversing maps with $y \le f(x)$ if and only if $x \le g(y)$; it is a monotone connection between $P$ and the dual of $Q$.