# An example with a non-increasing function

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$.

Let $\mathfrak{A}$ and $\mathfrak{B}$ are posets. A pointfree funcoid from $\mathfrak{A}$ to $\mathfrak{B}$ is a pair $\left( \alpha ; \beta \right)$ of functions $\alpha \in \mathfrak{B}^{\mathfrak{A}}$ and $\beta \in \mathfrak{A}^{\mathfrak{B}}$ such that $\alpha \left( x \right) \curlyvee y \Leftrightarrow \beta \left( y \right) \curlyvee x$ for every $x \in \mathfrak{A}$, $y \in \mathfrak{B}$.

I denote $\left\langle \left( \alpha ; \beta \right) \right\rangle = \alpha$ for every pointfree funcoid $\left( \alpha ; \beta \right)$.

Conjecture There exist a pointfree funcoid $f$ such that $\langle f \rangle$ is not an increasing function.

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What does it mean for a point free funcoid to be increasing? A point free funcoid is an order pair of functions $(\alpha, \beta)$. It is not clear what increasing would mean. Both are increasing? –  William Jul 21 '12 at 19:30
@William: In this my question I haven't said anything about "increasing pointfree funcoids". Instead, I ask whether the function $\langle f\rangle = \alpha$ is always increasing (=monotone). –  porton Jul 21 '12 at 19:51