Heyting Algebra - Properties - Regular and Complemented Elements

Regular and Complemented Elements

An element x of a Heyting algebra H is called regular if either of the following equivalent conditions hold:

The equivalence of these conditions can be restated simply as the identity, valid for all .

Elements and of a Heyting algebra are called complements to each other if and . If it exists, any such is unique and must in fact be equal to . We call an element complemented if it admits a complement. It is true that if is complemented, then so is, and then and are complements to each other. However, confusingly, even if is not complemented, may nonetheless have a complement (not equal to ). In any Heyting algebra, the elements 0 and 1 are complements to each other. For instance, it is possible that is 0 for every different from 0, and 1 if, in which case 0 and 1 are the only regular elements.

Any complemented element of a Heyting algebra is regular, though the converse is not true in general. In particular, 0 and 1 are always regular.

For any Heyting algebra H, the following conditions are equivalent:

1. H is a Boolean algebra;
2. every x in H is regular;
3. every x in H is complemented.

In this case, the element a → b is equal to ¬a ∨ b.

The regular (resp. complemented) elements of any Heyting algebra H constitute a Boolean algebra H_reg (resp. H_comp), in which the operations ∧, ¬ and →, as well as the constants 0 and 1, coincide with those of H. In the case of H_comp, the operation ∨ is also the same, hence H_comp is a subalgebra of H. In general however, H_reg will not be a subalgebra of H, because its join operation ∨_reg may be differ from ∨. For x, y ∈ H_reg, we have x ∨_reg y = ¬(¬x ∧ ¬ y). See below for necessary and sufficient conditions in order for ∨_reg to coincide with ∨.