- Thanks to Keith Kearnes for the replies ...

> Any Heyting algebra with an underlying

Thanks. I don't know why I didn't think

> loop (or quasigroup) structure must be Boolean.

of trying a 3-element Heyting algebra

before playing with this.

> ... one can show that there are exactly

That does not seem too surprising in view of

> 4 sets of operations {x+y, x*y, -x, 0, 1} such that

> (i) + and * are associative and * distributes over + on {0,a,1}, and

> (ii) these operations are the usual Boolean ring operations when

> restricted to {0,1}.

> But in none of the 4 cases do the operations generate

> all the Heyting operations on {0,a,1}.

> Therefore, "ring-like" cannot mean "semiring".

Steve Vickers' observation that reasonable

definitions of + in terms of the Heyting operations

are nonassociative.

I suspect that whatever these things are, they are

not part of a well-trodden variety. I also do not

have hopes that they have much structure.

Cheers,

Michael - At 10:35 31/01/2004 -0500, Michael Kinyon wrote:
>By the way, just so that I am sure we are all talking about

Yes, in my experience that seems to be widely understood as standard.

>the same things, the definition of Heyting algebra with

>which I am comfortable is a pseudo-complemented, distributive

>lattice with 0. ...

> ...(I think the distributivity is redundant.)

Yes, it is.

One scenic route to this is via category theory. A Heyting algebra is

exactly a cartesian closed lattice.

Cartesian closure says that for any object b, the functor b x - has a right

adjoint, namely (-)^b. In the Heyting algebra case, where product x is meet

/\, and exponentiation c^b is the Heyting arrow b->c, this follows directly

from the fact that

b/\a <= c iff a <= b->c

Since the functor b x - is a left adjoint, it preserves all colimits, which

in the lattice case means joins. To say that b /\ - preserves finite joins

is to say that /\ distributes over \/, and so the lattice is distributive.

Notice a bit more. b /\ - preserves all joins that exist. Hence if the

Heyting algebra is a complete lattice, then it has "frame distributivity" -

/\ distributes over arbitrary joins. (Unlike the finitary case, we can't

deduce from this that \/ distributes over all meets.)

Frames (frame distibutive complete lattices), which are studied in

point-free topology as an abstract lattice-theoretic version of topologies,

are often called "complete Heyting algebras". However, that is slightly

naughty since the different phrases imply different homomorphisms.

Regards,

Steve Vickers.