De Morgan algebra
Encyclopedia
In mathematics
, a De Morgan algebra 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
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De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic
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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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a De Morgan algebra is a structure A = (A, ∨, ∧, 0, 1, ¬) such that:
- (A, ∨, ∧, 0, 1) is a bounded distributive latticeDistributive latticeIn mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
, 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 noncontradictionLaw of noncontradictionIn classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...
)
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
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
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De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
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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.