Let $N\geq2.$ Let $F$ be a function from $\left\{ 0,1\right\} ^{N}$ to itself dreceasing for the product order defined by $$ (x_1,x_2,\ldots,x_N)\leq (y_1,\ldots,y_N)\ \text{ if and only if for all }i,\ x_i\leq y_i $$

Here, $F$ being decreasing means $$x\leq y \Rightarrow F(y)\leq F(x)$$

Suppose moreover that the $i^{th}$ component of $F$ does not depend on the $i^{th}$ variable.

Is it true that $F$ has a unique fixed point ?

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