Given posets $(X, \leq)$, and $(Y, \leq)$, an order morphism is a map $f : X \to Y$ such that $x \leq y$ implies $f(x) \leq f(y)$.
Given two order morphism $f: X \to Y$ and $g: Y \to X$, we say that $f$ is a left adjoint of $g$ (or that $g$ is a right adjoint of $f$) if for all $x \in X, y \in Y$ we have $$ x \leq g(y) \Leftrightarrow f(x) \leq y $$ in which case we also call $(f, g)$ an adjoint pair.
Classical examples are:
- For an extension of fields $F \subseteq K$, let $\mathcal{F}$ be the poset of intermediate fields (ordered by inclusion) and $\mathcal{G}$ the poset of subgroups of the Galois group Gal$(K/F)$, ordered by inclusion. Then $$ f: \mathcal{G} \to \mathcal{F}^{op} : H \mapsto \text{Fix}(H) \\ g: \mathcal{F}^{op} \to \mathcal{G} : L \mapsto \text{Gal}(K/L) $$ forms an adjoint pair.
- Let $(X, \leq)$ be a poset and for $Y \subseteq X$ define ub$(Y) = \lbrace x \in X \vert \forall y \in Y: y \leq x \rbrace$, similarly define lb$(Y) = \lbrace x \in X \vert \forall y \in Y: x \leq y \rbrace$. Letting $\mathcal{P}(X)$ be the power set of $X$, ordered by inclusion, then $$ \text{ub} : \mathcal{P}(X) \to \mathcal{P}(X)^{op} : Y \mapsto \text{ub}Y \\ \text{lb} : \mathcal{P}(X)^{op} \to \mathcal{P}(X) : Y \mapsto \text{lb}Y $$ forms an adjoint pair.
- For a field $k$ and the $n$-dimensional affine space $\mathbb{A}^{n}(k)$, the maps $$ V : \mathcal{P}(k[X_1, ..., X_n]) \to \mathcal{P}(\mathbb{A}^{n}(k))^{op} : I \mapsto \lbrace x \in \mathbb{A}^{n}(k) \vert \forall f \in I : f(x) = 0 \rbrace \\ I : \mathcal{P}(\mathbb{A}^{n}(k))^{op} \to \mathcal{P}(k[X_1, ..., X_n]) : V \mapsto \lbrace f \in k[X_1, ..., X_n] \vert \forall x \in V : f(x) = 0 \rbrace $$ form an adjoint pair.
All these examples, however, rely on arbitrarily considering the opposite relation on one of the two posets (for a poset $(X, \leq)$, the poset $(X^{op}, \leq^{op})$ is the set $X$ with order $x \leq^{op} y \Leftrightarrow y \leq x$). In fact, all maps considered 'reverse' the natural order and these structures are also known als general Galois connections; more examples are given in this question. Are there interesting examples of adjoint pairs which respect the natural order on both sets? I'm looking for adjoint pairs of posets, not of categories.