## RE: [univalg] Re: Heyting algebra / ring-like thing correspondence

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• Thanks to Keith Kearnes for the replies ... ... Thanks. I don t know why I didn t think of trying a 3-element Heyting algebra before playing with this. ...
Message 1 of 6 , Feb 1, 2004
Thanks to Keith Kearnes for the replies ...

> Any Heyting algebra with an underlying
> loop (or quasigroup) structure must be Boolean.

Thanks. I don't know why I didn't think
of trying a 3-element Heyting algebra
before playing with this.

> ... one can show that there are exactly
> 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".

That does not seem too surprising in view of
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
• ... Yes, in my experience that seems to be widely understood as standard. ... Yes, it is. One scenic route to this is via category theory. A Heyting algebra is
Message 2 of 6 , Feb 1, 2004
At 10:35 31/01/2004 -0500, Michael Kinyon wrote:
>By the way, just so that I am sure we are all talking about
>the same things, the definition of Heyting algebra with
>which I am comfortable is a pseudo-complemented, distributive
>lattice with 0. ...

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

> ...(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.
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