de Morgan algebra

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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, the laws
¬x ∨ x = 1 (law of the excluded middle), and
¬x ∧ x = 0 (law of noncontradiction)
do not always hold.

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If a De Morgan algebra additionally satisfies x โˆง ยฌx โ‰ค y โˆจ ยฌy,
it is called a Kleene algebra.[1][3]
(This notion should not to be confused with the other Kleene algebra
generalizing regular expressions.)

This notion has also been called a normal i-lattice by Kalman.