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Topic: Negation in a Bi-Heyting Lattice
Replies: 0

 Rock Brentwood Posts: 116 Registered: 6/18/10
Negation in a Bi-Heyting Lattice
Posted: Dec 20, 2013 5:41 PM

A lattice L can and its dual can both have the structure of a Heyting
lattice. Denote the bottom by 0, top by 1, join by |, meet by &. These
satisfy the properties:
x |- 1
0 |- x
x |- a&b iff x |- a, x |- b
a|b |- x iff a |- x, b |- x
where |- denotes the lattice ordering.
In a Heyting lattice one also has the conditional operatior a -> b and
negation !a = a -> 0 which satisfy the properties:
x |- a -> b iff x&a |- b
x |- !a iff x&a |- 0.
For a bi-Heyting lattice, one can also define the "unless" or
"exclusion" operator a - b with its own negation ~b = 1 - b, which
satisfy the properties:
a - b |- x iff a |- b | x
~b |- x iff 1 |- b | x.

So, the question is: what is the combined structure of produced by the
~ and ! operators (e.g. ~a |- !a, !!a |- a |- ~~a)? More particularly,
what is the free sublattice generated from an element a by the
combinations of ~ and !? Is it finite, in general or can it be
infinite?