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Re: A deMorgan law for Heyting lattices
Posted:
Aug 30, 2013 7:54 AM


Victor Porton <porton@narod.ru> wrote: > I have already asked this question in math.stackexchange.com: > http://math.stackexchange.com/questions/478886/ademorganlawforheytinglattices > > 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 UTFimpaired 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



