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Topic: A de-Morgan law for Heyting lattices
Replies: 9   Last Post: Aug 30, 2013 7:54 AM

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@less@ndro

Posts: 215
Registered: 12/13/04
Re: A de-Morgan law for Heyting lattices
Posted: Aug 30, 2013 7:54 AM
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Victor Porton <porton@narod.ru> wrote:
> I have already asked this question in math.stackexchange.com:
> http://math.stackexchange.com/questions/478886/a-de-morgan-law-for-heyting-lattices
>
> but the original version of my question was with an error.
>
> I've corrected the question and now ask you to help me finding the answer.
>
> The question:
>
> Please show that ¬(a?b)=¬a?¬b for any Heyting algebra.


[ for the benefit of our UTF-impaired readers I use ~, v and & instead
of ¬, ? and ? ]

The basic observation is the equivalence

x <= ~y <==> x & y <= 0 <==> y & x <= 0 <==> y <= ~x

in a Heyting lattice (H, <= ). This shows that ~ : H --> H^op is
left adjoint to ~ : H^op --> H and therefore preserves joins.

Of course you can also use Yoneda to calculate directly: given a, b in H,
then for every x in H you have

x <= ~(a v b)

<==> x & (a v b) <= 0

<==> (x & a) v (x & b) <= 0

<==> x & a <= 0 and x & b <= 0

<==> x <= ~a and x <= ~b

<==> x <= (~a & ~b)

--
Marc



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