Know the Boolean algebras order for Hasse Diagram

Question

I have (F(B^2,B)) for the functions B^2 in B, defined by f#g -> f(x)<=g(x) for each x in B^2.

I need to draw the hasse diagram for the boolean algebra (F(B^2,B),#), but don't know if is a order 3 like this: Hasse or a order 2.

How can i know it?

Answer 1

If I understood the question correctly, you need to draw a Hasse diagram of the poset $(F(B^2, B), \#)$, where $f\#g \Leftrightarrow (\forall x \in B^2)(f(x) \leqslant g(x))$.

Every function $f \colon B^2 \to B$ is uniquely determined by its vector of values $$\hat{f} = (f(0, 0), f(0, 1), f(1, 0), f(1, 1)) \in B^4,$$ (and conversely, every vector in $B^4$ defines such a function).

So your poset is the same as $(B^4, \leqslant)$, where $$(x_0, x_1, x_2, x_3) \leqslant (y_0, y_1, y_2, y_3) \Leftrightarrow (\forall i \in \{0, 1, 2, 3\})(x_i \leqslant y_i).$$ Its Hasse diagram is a $4$-dimensional Boolean cube:

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