De Morgan algebra

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Definition
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.
It should be noted that:
¬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.

Ref Injective de Morgan and Kleene Algebras Roberto Cignoli Proceedings of the American Mathematical Society, Vol. 47, No. 2 (Feb., 1975), pp. 269-278