In mathematics, a De Morgan algebra is a structure A = (A, ∨, ∧, 0, 1, ¬) such that:
In a De Morgan algebra:
do not always hold (when they do, the algebra becomes a Boolean algebra).
Remark: It follows that ¬( x∨y) = ¬x∧¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1∨0 = ¬1∨¬¬0 = ¬(1∧¬0) = ¬¬0 = 0). Thus ¬ is a dual automorphism.
De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic.
The standard fuzzy algebra F = ([0, 1], max(x, y), min(x, y), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.