A Heyting lattice is a Brouwerian lattice with a bottom element . Equivalently, is Heyting iff it is relatively pseudocomplemented and pseudocomplemented iff it is bounded and relatively pseudocomplemented.
Let denote the pseudocomplement of and the pseudocomplement of relative to . Then we have the following properties:
(equivalence of definitions)
(if , then by the definition of .)
iff ( implies that whenever . In particular , so or . On the other hand, if , then .)
and (already true in any pseudocomplemented lattice)
If , then so , and likewise, so . This means precisely that . ∎
(since and )
Note that in property 4, , whereas is in general not true, contrasting with the equality in a Boolean lattice, where is the complement operator. It is easy to see that if for all in a Heyting lattice , then is a Boolean lattice. In this case, the pseudocomplement coincides with the complement of an element , and we have the equality in property 7: , meaning that the concept of relative pseudocomplementation (http://planetmath.org/RelativelyPseudocomplemented) coincides with the material implication in classical propositional logic.
A Heyting algebra is a Heyting lattice such that is a binary operator on . A Heyting algebra homomorphism between two Heyting algebras is a lattice homomorphism that preserves , and . In addition, if is a Heyting algebra homomorphism, preserves psudocomplementation: .
In the literature, the assumption that a Heyting algebra contains is sometimes dropped. Here, we call it a Brouwerian lattice instead.
|Date of creation||2013-03-22 16:33:03|
|Last modified on||2013-03-22 16:33:03|
|Last modified by||CWoo (3771)|