International Conference Logic, Algebra and Truth Degrees [LATD2008]

1. International ConferenceLogic, Algebra and Truth Degrees [LATD2008] ___________________________________________ UNIVERSITA DEGLI STUDI DI SIENA Santa Chiara College 8-11 September 2008, Siena, Italy ___________________________________________ Organizer: Working Group on Mathematical Fuzzy Logic ___________________________________________

2. On the logic of approximation problem solving Mircea Sularia Polytechnic University of Bucharest, Faculty of Applied Sciences, Department of Mathematics 2, Romania

3. 1. Symmetric Boolean algebras • The notion of symmetric Boolean algebra has been introduced in 1954 by Moisil in connection with the study of electrical circuits with valves. • A symmetric Boolean algebra (SB-algebra) is a Boolean algebra equipped with an involutive automorphism.

4. 2. Involutive Boolean algebras • Involutive Boolean algebra is a term introduced also by Moisil in connection with its notion of symmetric Boolean algebra. • An involutive Boolean algebra (IB-algebra) is a Boolean algebra together with an involutive dual automorphism.

5. 3. SB-algebras and IB-algebras are isomorphic classes Symmetric Boolean algebras and involutive Boolean algebras are isomorphic classes. This fact will be expressed in the sequel.

6. Isomorphism SB IB

7. Structures including involutive Boolean algebras – p1 • Involutive De Morgan algebras • Quasi-topological square lattices • Brouwerian D-algebras and Involutive Brouwerian D-algebras (BrD-algebra and IBrD-algebra)

8. Structures including involutive Boolean algebras – p2 • Congruences in BrD-algebras • An equational definition and connected HBr-pairs • Normal filters in IBrD-algebras

9. Structures including involutive Boolean algebras – p3 [Logic-theoretical motivation] • Symmetric Kolmogorov logic • Gödel-Dummett Kolmogorov logic • Completeness theorems with respect to IBrD-algebras and Stone-Heyting algebras

10. Abstract of communicationPartially ordered structures including involutive Boolean algebras

11. Connected SHMV-pairs • A Stone-Heyting MV-pair (StH_MV,StBr_MV) is called connected if a connection is given. • Connection is a pair (f, g) of bijective functions such that • f : StH  StBr is an isomorphism from StH_MV onto the dual d(StBr_MV) • g : StBr  StH is an isomorphism from StBr_MV onto the dual d(StH_MV) • f o g = id[StBr] and g o f = id[StH]

12. Stone-Heyting MVD-algebra defined by connected SHMV-pairs • Let Cp = [(StH_MV,StBr_MV), (f,g)] be a connected SHMV-pair of algebras with respect to a connection (f, g) • A standard structure of IBrDMV-algebra denoted by A[Cp] exists on the direct product set StH x StBr obtained by a similar construction realized for IBrD-algebras associated with connected HBr-pairs. • Theorem. For any IBrDMV-algebra A, there exists an embedding of A into an algebra A[Cp] defined by a connected SHMV-pair Cp.