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Negation in a BiHeyting 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 ab  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 biHeyting 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?



