De Morgan algebra

In mathematics, a De Morgan algebra 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 = ([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.

References

• "Injective de Morgan and Kleene Algebras", Roberto Cignoli, Proceedings of the American Mathematical Society, Vol. 47, No. 2 (Feb., 1975), pp. 269–278
• Thomas Scott Blyth; J. C. Varlet (1994). Ockham algebras. Oxford University Press. ISBN 9780198599388.