Formal Definition
A Heyting algebra is a bounded lattice such that for all and in there is a greatest element of such that
This element is the relative pseudo-complement of with respect to, and is denoted . We write 1 and 0 for the largest and the smallest element of, respectively.
In any Heyting algebra, one defines the pseudo-complement of any element by setting . By definition, and is the largest element having this property. However, it is not in general true that, thus is only a pseudo-complement, not a true complement, as would be the case in a Boolean algebra.
A complete Heyting algebra is a Heyting algebra that is a complete lattice.
A subalgebra of a Heyting algebra is a subset of containing 0 and 1 and closed under the operations and . It follows that it is also closed under . A subalgebra is made into a Heyting algebra by the induced operations.
Read more about this topic: Heyting Algebra
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