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

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  • Kinyon, Michael K.
    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
    • S Vickers
      ... 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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