Editing Heyting algebra (section)

===Regular and complemented elements===
An element ''x'' of a Heyting algebra ''H'' is called '''regular''' if either of the following equivalent conditions hold:
#''x'' =  ¬¬''x''.
#''x'' =  ¬''y'' for some ''y'' in ''H''.
The equivalence of these conditions can be restated simply as the identity ¬¬¬''x'' = ¬''x'', valid for all ''x'' in ''H''.

Elements ''x'' and ''y'' of a Heyting algebra ''H'' are called '''complements''' to each other if ''x''∧''y'' = 0 and ''x''∨''y'' = 1. If it exists, any such ''y'' is unique and must in fact be equal to ¬''x''. We call an element ''x'' '''complemented''' if it admits a complement. It is true that ''if'' ''x'' is complemented, then so is ¬''x'', and then ''x'' and ¬''x'' are complements to each other. However, confusingly, even if ''x'' is not complemented, ¬''x'' may nonetheless have a complement (not equal to ''x''). In any Heyting algebra, the elements 0 and 1 are complements to each other. For instance, it is possible that ¬''x'' is 0 for every ''x'' different from 0, and 1 if ''x'' = 0, 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:
# ''H'' is a [[Boolean algebra (structure)|Boolean algebra]];
# every ''x'' in ''H'' is regular;<ref>Rutherford (1965), Th.26.2 p.78.</ref>
# every ''x'' in ''H'' is complemented.<ref>Rutherford (1965), Th.26.1 p.78.</ref>
In this case, the element {{nowrap|1=''a''→''b''}} is equal to {{nowrap|1=¬''a'' ∨ ''b''.}}

The regular (respectively complemented) elements of any Heyting algebra ''H'' constitute a Boolean algebra ''H''<sub>reg</sub> (respectively ''H''<sub>comp</sub>), in which the operations ∧, ¬ and →, as well as the constants 0 and 1, coincide with those of ''H''. In the case of ''H''<sub>comp</sub>, the operation ∨ is also the same, hence ''H''<sub>comp</sub> is a subalgebra of ''H''. In general however, ''H''<sub>reg</sub> will not be a subalgebra of ''H'', because its join operation ∨<sub>reg</sub> may be different from ∨. For {{nowrap|1=''x'', ''y'' ∈ ''H''<sub>reg</sub>,}} we have {{nowrap|1=''x'' ∨<sub>reg</sub> ''y'' = ¬(¬''x'' ∧ ¬''y'').}} See below for necessary and sufficient conditions in order for ∨<sub>reg</sub> to coincide with ∨.