De Morgan algebra

English

Alternative forms

• de Morgan algebra

Etymology

Named after British mathematician and logician Augustus De Morgan (1806–1871). The notion was introduced by Grigore Moisil.

Noun

De Morgan algebra (plural De Morgan algebras)

1. (algebra, order theory) A bounded distributive lattice equipped with an involution (typically denoted ¬ or ~) which satisfies De Morgan's laws.
   • 1980, H. P. Sankappanavar, “A Characterization of Principal Congruences of De Morgan Algebras and its Applications”, in A. I. Arruda, R. Chuaqui, N. C. A. Da Costa, editors, Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, page 341:

     Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice.

   • 2000, Luo Congwen, Topological De Morgan Algebras and Kleene-Stone Algebras: The Journal of Fuzzy Mathematics, Volume 8, Pages 1-524, page 268:

     By a topological de Morgan algebra we shall mean an abstract algebra $(A,\land ,\lor ,\ ',l)$ where $(A,\land ,\lor ,l)$ is a de Morgan algebra,

   • 2009, George Rahonis, “Chapter 12: Fuzzy Languages”, in Manfred Droste, Werner Kuich, Heiko Vogler, editors, Handbook of Weighted Automata, Springer, page 486:

     If $(L,\leq ,{^{-}})$ is a bounded distributive lattice with negation function (resp. a De Morgan algebra), then $(L\langle \!\langle S\rangle \!\rangle ,\leq ,{^{-}})$ constitutes also a bounded distributive lattice with negation function (resp. a De Morgan algebra); for every $r\in L\langle \!\langle S\rangle \!\rangle$ its negation ${\overline {r}}\in L\langle \!\langle S\rangle \!\rangle$ is defined by $({\overline {r}},s)={\overline {(r,s)}}$ for every $s\in S$.

Hypernyms

• Ockham algebra
  • distributive lattice

Hyponyms

• Kleene algebra
  • Boolean algebra

Translations

bounded distributive lattice equipped with an involution satisfying De Morgan's laws

Further reading

• De Morgan's laws on Wikipedia.Wikipedia
• Complemented lattice on Wikipedia.Wikipedia