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, the laws
- ¬x ∨ x = 1 (law of the excluded middle), and
- ¬x ∧ x = 0 (law of noncontradiction)
do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them 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 of (A, ∨, ∧, 0, 1).
If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that:
- (A, ≤) is a bounded distributive lattice, and
- ¬¬x = x, and
- x ≤ y → ¬y ≤ ¬x.
De Morgan algebras were introduced by Grigore Moisil[1][2] around 1935,[2] although without the restriction of having a 0 and a 1.[3] They were then variously called quasi-boolean algebras in the Polish school, e.g. by Rasiowa and also distributive i-lattices by J. A. Kalman.[2] (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.[1][2]
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.
Another example is Dunn's four-valued semantics for De Morgan algebra, which has the values T(rue), F(alse), B(oth), and N(either), where
- F < B < T,
- F < N < T, and
- B and N are not comparable.[2]
- ^ a b Blyth, T. S.; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. pp. 4–5. ISBN 978-0-19-859938-8.
- ^ a b c d e Béziau, Jean-Yves (2012). "A History of Truth-Values". In Gabbay, Dov M.; Pelletier, Francis Jeffry; Woods, John (eds.). Logic: A History of its Central Concepts. North Holland (an imprint of Elsevier). pp. 280–281. ISBN 978-0-08-093170-8.
- ^ Cignoli, Roberto (1975). "Injective de Morgan and Kleene Algebras" (PDF). Proceedings of the American Mathematical Society. 47 (2): 269–278. doi:10.1090/S0002-9939-1975-0357259-4. JSTOR 2039730.