In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that:
- (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and
- ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies De Morgan's laws)
In a De Morgan algebra:
- ¬x ∨ x = 1 (law of the excluded middle), and
- ¬x ∧ x = 0 (law of noncontradiction)
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 = (, 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.
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