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
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)