| Area of Mathematics: | Algebra & Geometry (EAG) | ||
| Semester: | 8ο | ||
| Course ID: | 82202 | ||
| Course Type: | Elective | ||
| Teaching hours per week: | Theory: 4 | Practice: 0 | Laboratory: 0 |
| ECTS : | 5 | ||
| Eclass: | |||
| Instructors: | Vaia Prassa |
Description
- Preorder and partial order structures, semilatitices and lattices, complete lattices. Axioms for (semi)lattices. Lattice extensions, residuated lattices, FL algebras and axiomatic extensions (BCI, BCK, BCW algebras). Heyting algebras (BCKW) and Boole algebras and rings. Normal extensions. Existence and uniqueness of normal extensions. Stone duality, Jonsson-Tarski theorem for Boole algebras with tensor and consequences of the theorem in distributive and non-distributive lattices with tensors.
- The Lindenbaum-Tarski construction for classic propositional logic. Classic propositional logic and Boole algebras. Extension of the construction in intuitionistic logic. Intuitionistic logic and Heyting algebras. Extension of the construction in infrastutural logic systems and conjugate lattices. Algebraic and set-theoretic interpretation through lattice representation.
- Generalization of the Lindenbaum-Tarsk construction, the algebrification concept and the logic system algebrification problem. The Leibniz tensor and the Blok-Pigozzi theorem. Relevance Logic systems as examples of non-algrebrificable systems. The relation between algebras classes and families of logic systems – Bridging and translation results. Basic cases – algebraic equivalents of the deduction theorem, Craig’s interpolation theorem and Beth’s definability theorem.
- Closure tensors, logically consistent tensors. Logic matrices. Matrix semantics for logic systems. Logic matrices deriven logic systems.
- Classification of logic systems: Introduction to Leibniz and Frege hierarchies.
Bibliography
- Josep Maria Font. Abstract Algebraic Logic, Studies in Logic, vol 60, College Publications, London, 2016.
- Galatos, P. Jipsen, T. Kowalski and H. Ono. Residuated Lattices: An algebraic glimpse at substructural logics. Studies in Logic and the Foundations of Mathematics, vol 151, Elsevier 2007.
- Birkhoff G., Lattice Theory, 3rd ed., Amer. Soc., 1967.
Davey B.A., Priestley H.A., Introduction to Lattices and Order, Cambridge Univ. Press, 1990.
