I'm just starting out a study of combinatorics using the Rota's text: Combinatorics: The Rota Way.
I'm confused about what I'm being asked to do here. We are given some binary operation ''implies'' which is defined:
$$x \implies y= x^c \vee y$$
The question is to use this map and $\hat{0}$ (which to my vague knowledge is a ''minimum'' in some sense) to generate the operations $\vee,\wedge$ and the element $\hat{1}$. Then I'm to give a set of axioms using $\implies$ and $\hat{0}$.
Ok so the only thing that makes sense to write is:
$$\hat{0} \implies {\hat{0}}=\hat{0}^c \vee \hat{0} = \hat{1}$$ (please confirm). So I have $\hat{1}$ but now how do I generate the other operations?
Edit; Thanks to comments I now see how to construct $\wedge$, $\vee$, $^c$. I'm still struggling with ''give a set of axioms using $\implies$ & $\hat{0}$. Am I supposed to just go through all the Lattice axioms, distribivity, etc and translate them? For example I tried to recreate idempotency by;
$$(x \implies 0) \implies x =x \implies (x \implies 0) \implies 0=x $$
Is this the right approach?