complete Heyting algebra
A Heyting algebra that is also a complete lattice is called a complete Heyting algebra. In the following, we give a lattice characterization of complete Heyting algebras without the relative pseudocomplementation operator .
Let be a complete Heyting algebra, then
for any , and
where , for any , and any subset of .
We first prove the identity . For any , we have iff iff for all iff for all iff , hence .
Next, we show . For any , we have iff iff iff . ∎
The converse of the above is also true.
Let be a complete lattice such that
where , for any , and any subset of . Then for any , defining
turns into a complete Heyting algebra.
We want to show that iff for any : iff . So ∎
From this, one readily concludes that any finite distributive lattice is Heyting.
|Title||complete Heyting algebra|
|Date of creation||2013-03-22 19:31:42|
|Last modified on||2013-03-22 19:31:42|
|Last modified by||CWoo (3771)|