De Morgan algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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:

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(xy), min(xy), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.

References[edit]

  • "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 978-0-19-859938-8.