Lindenbaum–Tarski algebra
- Lindenbaum–Tarski algebra
In mathematical logic, the Lindenbaum-Tarski algebra "A" of a logical theory "T" consists of the equivalence classes of sentences "p" of the theory, under the equivalence relation ~ defined by
:"p" ~ "q" when "p" and "q" are logically equivalent in "T".
That is, in "T" the sentence "q" can be deduced from "p", and "p" from "q".
Operations in "A" are inherited from those available in "T", typically conjunction and disjunction, where they are well-defined on the classes. When negation is also present in "T", then "A" is a Boolean algebra, provided the logic is classical. Conversely, for every Boolean algebra "A", there is a theory "T" of (classical) sentential logic such that the Lindenbaum-Tarski algebra of "T" is isomorphic to "A". In other words, every Boolean algebra is (up to isomorphism) a Lindenbaum-Tarski algebra.
Heyting algebra and interior algebra are the Lindenbaum-Tarski algebras for intuitionistic logic and the modal logic S4, respectively.
Sometimes called simply Lindenbaum algebra, this construction is named for logicians Adolf Lindenbaum and Alfred Tarski.
ee also
*Abstract algebraic logic
*Algebraic logic
*List of Boolean algebra topics
References
*cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | id = ISBN 1-568-81262-0
Wikimedia Foundation.
2010.
Look at other dictionaries:
Lindenbaum — is a surname, meaning Tilia in German; the nearest British tree name is Lime tree. It may refer to:* Adolf Lindenbaum, Polish mathematician ** Lindenbaum s lemma ** Lindenbaum–Tarski algebra * John Lindenbaum, musician * Der Lindenbaum one of the … Wikipedia
Alfred Tarski — Infobox scientist name = Alfred Tarski caption = birth date = birth date|1901|01|14 birth place = Warsaw, Poland (under Russian rule at the time) death date = death date|1983|10|26 death place = Berkeley, California fields = Mathematics, logic,… … Wikipedia
Boolean algebra (logic) — For other uses, see Boolean algebra (disambiguation). Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of… … Wikipedia
Interior algebra — In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and… … Wikipedia
List of Boolean algebra topics — This is a list of topics around Boolean algebra and propositional logic. Contents 1 Articles with a wide scope and introductions 2 Boolean functions and connectives 3 Examples of Boolean algebras … Wikipedia
Adolf Lindenbaum — (June 12, 1904 in Warsaw, Russian Empire (now Poland) – 1941 in Paneriai), was a Polish logician and mathematician.He was a student of Wacław Sierpiński, became a distinguished author of works on set theory and had served as an Assistant… … Wikipedia
MV-algebra — In abstract algebra, a branch of pure mathematics, an MV algebra is an algebraic structure with a binary operation oplus, a unary operation eg, and the constant 0, satisfying certain axioms. MV algebras are models of Łukasiewicz logic; the… … Wikipedia
Adolf Lindenbaum — (* 12. Juni 1904 in Warschau; † September 1941 in Paneriai) war ein polnischer Logiker und Mathematiker jüdischer Abstammung, der mit der Lemberg Warschau Schule verbunden war. Er studierte Mathematik bei Wacław Sierpiński, seine… … Deutsch Wikipedia
Tarski — Alfred Tarski in Berkeley Alfred Tarski bzw. ursprünglich Alfred Teitelbaum (* 14. Januar 1901 (nach anderen Quellen: 1902) in Warschau; † 26. Oktober 1983 in Berkeley, USA) war ein polnisch US amerikanischer … Deutsch Wikipedia
Boolean algebra (structure) — For an introduction to the subject, see Boolean algebra#Boolean algebras. For the elementary syntax and axiomatics of the subject, see Boolean algebra (logic). For an alternative presentation, see Boolean algebras canonically defined. In abstract … Wikipedia
