# Describe which partial orderings yield boolean algebras

I thougt about propositional logic and boolean algebras and how propositional logic is (at least from one point of view) not really about $\land,\lor,\neg,...$ but about boolean operators, i.e. n-ary functions which map $0$-$1$-sequences of length $n$ into $\{0,1\}$, and how to use a finite subset of those (base) to encode all of them.

With boolean algebras it's a little bit similar, one way to think of them is as a mathematical structure associated with a set $X\supseteq\{0,1\}$ which enlarges each boolean operator $f$ to an $X$-valued operator ${\cal X}(f)$ such that this operation is compatible with function composition, i.e. $${\cal X}(f)({\cal X}(g_1),\dots,{\cal X}(g_n))={\cal X}(f(g_1,\dots,g_n)).$$ But any boolean algebra is also a lattice (via $\land$ and $\lor$) and any lattice structure can be described as partial order. I just wonder if it's also possible to describe the partial order on a boolean algebra in terms which do not prioritize the operations $\land$ and $\lor$.

$$\neg x = (x \Rightarrow \bot)$$
where $\Rightarrow$ denotes implication / exponential and $\bot$ denotes false / the initial object. There is a natural map $x \to \neg \neg x$, and a Heyting algebra is a Boolean algebra iff this natural map is always an isomorphism, or in other words, iff double negation holds.