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De Morgan algebra
System of logic lacking the excluded middle law

In mathematics, a De Morgan algebra, named after Augustus De Morgan, is a bounded distributive lattice with an involution ¬ satisfying De Morgan's laws (De Morgan's laws) and the condition ¬¬x = x. Unlike Boolean algebras, the law of excluded middle and law of noncontradiction do not necessarily hold. Introduced by Grigore Moisil around 1935, De Morgan algebras are important in fuzzy logic where truth values range continuously, such as in the standard fuzzy algebra [0,1]. Another example is Dunn's four-valued semantics, featuring truth values T, F, B, and N, illustrating the flexibility of De Morgan structures beyond classical logic.

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Kleene algebra

If a De Morgan algebra additionally satisfies x ∧ ¬x ≤ y ∨ ¬y, it is called a Kleene algebra.910 (This notion should not be confused with the other Kleene algebra generalizing regular expressions.) This notion has also been called a normal i-lattice by Kalman.

Examples of Kleene algebras in the sense defined above include: lattice-ordered groups, Post algebras and Łukasiewicz algebras.11 Boolean algebras also meet this definition of Kleene algebra. The simplest Kleene algebra that is not Boolean is Kleene's three-valued logic K3.12 K3 made its first appearance in Kleene's On notation for ordinal numbers (1938).13 The algebra was named after Kleene by Brignole and Monteiro.14

De Morgan algebras are not the only plausible way to generalize Boolean algebras. Another way is to keep ¬x ∧ x = 0 (i.e. the law of noncontradiction) but to drop the law of the excluded middle and the law of double negation. This approach (called semicomplementation) is well-defined even for a (meet) semilattice; if the set of semicomplements has a greatest element it is usually called pseudocomplement, and the resulting algebra is a Heyting algebra. If the pseudocomplement satisfies the law of the excluded middle, the resulting algebra is also Boolean. However, if only the weaker law ¬x ∨ ¬¬x = 1 is required, this results in Stone algebras.15 More generally, both De Morgan and Stone algebras are proper subclasses of Ockham algebras.

See also

Further reading

  • Balbes, Raymond; Dwinger, Philip (1975). "Chapter XI. De Morgan Algebras and Lukasiewicz Algebras". Distributive lattices. University of Missouri Press. ISBN 978-0-8262-0163-8.
  • Birkhoff, G. (1936). "Reviews: Moisil Gr. C.. Recherches sur l'algèbre de la logique. Annales scientifiques de l'Université de Jassy, vol. 22 (1936), pp. 1–118". The Journal of Symbolic Logic. 1 (2): 63. doi:10.2307/2268551. JSTOR 2268551.
  • Batyrshin, I.Z. (1990). "On fuzzinesstic measures of entropy on Kleene algebras". Fuzzy Sets and Systems. 34 (1): 47–60. doi:10.1016/0165-0114(90)90126-Q.
  • Kalman, J. A. (1958). "Lattices with involution" (PDF). Transactions of the American Mathematical Society. 87 (2): 485–491. doi:10.1090/S0002-9947-1958-0095135-X. JSTOR 1993112.
  • Pagliani, Piero; Chakraborty, Mihir (2008). A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer Science & Business Media. Part II. Chapter 6. Basic Logico-Algebraic Structures, pp. 193-210. ISBN 978-1-4020-8622-9.
  • Cattaneo, G.; Ciucci, D. (2009). "Lattices with Interior and Closure Operators and Abstract Approximation Spaces". Transactions on Rough Sets X. Lecture Notes in Computer Science 67–116. Vol. 5656. pp. 67–116. doi:10.1007/978-3-642-03281-3_3. ISBN 978-3-642-03280-6.
  • Gehrke, M.; Walker, C.; Walker, E. (2003). "Fuzzy Logics Arising From Strict De Morgan Systems". In Rodabaugh, S. E.; Klement, E. P. (eds.). Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets. Springer. ISBN 978-1-4020-1515-1.
  • Dalla Chiara, Maria Luisa; Giuntini, Roberto; Greechie, Richard (2004). Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics. Springer. ISBN 978-1-4020-1978-4.

References

  1. Blyth, T. S.; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. pp. 4–5. ISBN 978-0-19-859938-8. 978-0-19-859938-8 ↩

  2. 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. 978-0-08-093170-8 ↩

  3. 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. 978-0-08-093170-8 ↩

  4. 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. https://www.ams.org/journals/proc/1975-047-02/S0002-9939-1975-0357259-4/S0002-9939-1975-0357259-4.pdf ↩

  5. 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. 978-0-08-093170-8 ↩

  6. Blyth, T. S.; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. pp. 4–5. ISBN 978-0-19-859938-8. 978-0-19-859938-8 ↩

  7. 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. 978-0-08-093170-8 ↩

  8. 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. 978-0-08-093170-8 ↩

  9. Blyth, T. S.; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. pp. 4–5. ISBN 978-0-19-859938-8. 978-0-19-859938-8 ↩

  10. 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. https://www.ams.org/journals/proc/1975-047-02/S0002-9939-1975-0357259-4/S0002-9939-1975-0357259-4.pdf ↩

  11. 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. https://www.ams.org/journals/proc/1975-047-02/S0002-9939-1975-0357259-4/S0002-9939-1975-0357259-4.pdf ↩

  12. Kaarli, Kalle; Pixley, Alden F. (21 July 2000). Polynomial Completeness in Algebraic Systems. CRC Press. pp. 297–. ISBN 978-1-58488-203-9. 978-1-58488-203-9 ↩

  13. Kleene, S. C. (1938). "On Notation for Ordinal Numbers". The Journal of Symbolic Logic. 3 (4): 150–155. doi:10.2307/2267778. JSTOR 2267778. /wiki/Stephen_Cole_Kleene ↩

  14. Brignole, D.; Monteiro, A. (1964). "Caracterisation des algèbres de Nelson par des egalités". Notas de Logica Matematica. 20. Instituto de Matematica Universidad del sur Bahia Blanca. A (possibly abbreviated) version of this paper appeared later in Proceedings of the Japan Academy: Brignole, Diana; Monteiro, Antonio (1967). "Caracterisation des algèbres de Nelson par des egalités, I". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 43 (4). doi:10.3792/pja/1195521624, Brignole, Diana; Monteiro, Antonio (1967). "Caracterisation des algèbres de Nelson par des egalités, II". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 43 (4). doi:10.3792/pja/1195521625. /wiki/Antonio_Monteiro_(mathematician) ↩

  15. Blyth, T. S.; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. pp. 4–5. ISBN 978-0-19-859938-8. 978-0-19-859938-8 ↩