Boolean Algebras and Logic
نویسنده
چکیده
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. Some special classes of Boolean algebras are also concerned with and, in particular, the relations of Boolean algebras to propositional logic and to set theory are studied in more details. We mention here that all what we do for propositional logic can also be extended to first order logic, yet we will not go for it in this paper.
منابع مشابه
Filter theory in MTL-algebras based on Uni-soft property
The notion of (Boolean) uni-soft filters in MTL-algebras is introduced, and several properties of them are investigated. Characterizations of (Boolean) uni-soft filters are discussed, and some (necessary and sufficient) conditions for a uni-soft filter to be Boolean are provided. The condensational property for a Boolean uni-soft filter is established.
متن کاملSemi-Cohen Boolean Algebras
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.
متن کاملRepresentations as Full Peirce Algebras
Extended Abstract In this Note I prove a representation theorem for Peirce algebras introduced in Brink, Britz and Schmidt (1994b, 1994a). I show that the class of full Peirce algebras is characterised by the class of complete and atomic Peirce algebras in which the set of relational atoms is restricted by two conditions. One requires simplicity. The other requires that each relational element ...
متن کاملar X iv : 0 80 9 . 05 38 v 1 [ m at h . L O ] 3 S ep 2 00 8 BOOLEAN ALGEBRAS AND LOGIC
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone’s theorem from Boolean algebra. We mention here that the method we used can also be e...
متن کاملBoolean Algebras in Visser Algebras
We generalize the double negation construction of Boolean algebras in Heyting algebras, to a double negation construction of the same in Visser algebras (also known as basic algebras). This result allows us to generalize Glivenko’s Theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras. Mathematics Subject Classification: P...
متن کامل