I would like to know whether a Heyting algebra gives rise to ring in a similar way that a Boolean algebra gives rise to a Boolean ring. In a Boolean algebra $(B,\lor,\land,\lnot,0,1)$ I can define multiplication and addition as follows:
$$a * b = a \land b$$ $$a + b = (a \land \lnot b) \lor (b \land \lnot a)$$
And I get a Boolean ring $(B,+,*,0,1)$. From the Boolean ring we can reconstruct the Boolean algebra again via the following definitions (right?):
$$a \land b = a * b$$ $$a \lor b = (a + 1) * (b + 1) + 1$$
In a Heyting algebra $(H,\rightarrow,\land,0,1)$ we do only have a pseudo complement. Can we nevertheless define a ring, that in turn would allow us to reconstruct the algebra?