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**Definition 3.5**

*A*

**Heyting Algebra H**is a**relative pseudo complemented distributive lattice**.**distributive**means that the following equations are satisfied for any

The property of being

**relative pseudo complemented lattice**means that for any two elements there exists a third element , such that:

*pseudo complement*of relative to i.e., the greatest element of the set , and it is denoted as .

A particular feature of the Heyting algebra is the negation operation. The negation of an element S is defined to be the pseudo-complement of S i.e. , therefore we can write

The above equation entails that is the least upper bound of the set , i.e. the biggest set that does not contain any element of S. From the above definition of negation operation it follows that the Heyting algebra does not satisfy the law of excluded middle, i.e. given any element S of an Heyting algebra, we have the following relation: .

*Proof*. Let us consider . This represents the least upper bound of S and therefore, given any other element in the Heyting algebra such that and then, . But, since for any S we have and it follows that .

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**Previous:**Functors Cecilia Flori 2007-01-02