2 Feb 2016 Tuesday |
Lecture 9:00-11:00 B0. 203 |
| Definitions of lattices. The equivalence of the two definitions.
Distributive lattices (Section 1-3, Ch. 1 in Univ. Alg. , 2.1-2.6, 2.8-2.14, 4.4, 4.10 in Lat and Ord. We did not prove 4.10). |
4 Feb 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Distributive and modular lattices, complete lattices, Boolean lattices and Boolean algebras (Section 3 in Univ Alg,
we didn't prove Theorems 3.5 and 3.6., and Sections 4.13 - 4.18 in Lat and Ord).
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4 Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here TUT 1.
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9 Feb 2016 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Heyting algebras, equational definition, infinite distributive law, linear Heyting algebras, Heyting algebras of up-sets of a poset.
(See Section 2.2.1 in here.)
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11 Feb 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Heyting algebras of open sets of a topological space, interior algebras, topological insight on Goedel's embedding, Boolean algebra of regular open elements of a Heyting algebra, Glivenko's theorem.
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11 Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here TUT 2.
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16 Feb 2016 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Congruences, homomorphic images, ideals and filters, maximal, prime and ultrafilters of Boolean algebras (Sections 6.1-6.10, 2.20-2.21, 10.7-10.12 in Lat and Ord). Note that in the lectures we worked mostly with filters, whereas Lat and Ord works mostly with ideals.
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18 Feb 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Prime filter theorem, Stone representation theorem (Sections 10.15-10.18, 10.20-10.22, 11.1-11.4 in Lat and Ord)
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18 Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here TUT 3.
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23 Feb 2016 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Stone duality (Sections 11.1-11.6).
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25 Feb 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Stone duality, Alexandroff and Stone-Cech compactifications of natural numbers, Priestley duality. (Sections 11.7 - 11.10, 11.17 - 11.27 in Lat & Ord). Note that in Lat and Ord Stone spaces are called Boolean spaces.
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25 Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
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The tutorial exercises can be found here TUT 4.
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1 March 2015 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Priestley duality, Esakia spaces. (Sections 11.18 - 11.32 in Lat & Ord,
check also the notes of Pat Morandi on duality in lattice theory, Sections 3 -5.)
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3 March 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
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Universal algebras, H, S and P (Ch 2. Sections 1-2, Univ. Alg.), subdirectly irreducible algebras, varieties (Ch 2. Sections 8-9, Univ. Alg.), Birkhoff's variety theorem.
Morphisms between Esakia spaces, Esakia duality, the correspondence between congruences and closed sets for distributive latices,
the correspondence between congruences and closed up-sets for Heyting algebras,
Subdirectly irreducible Boolean algebras, distributive lattices and Heyting algebras.
Sections 2.3.2 - 2.3.4 here, Sections 11.27 - 11.32 in Lat and Ord, consult also Morandi's notes.
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3 March 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here
TUT 5.
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8 March 2015 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Algebraic completeness of classical and intuitionistic logics (Sections 11.11 - 11.16 in Lat & Ord,
consult also Section 4.3 in Notes on intuitionistic logic, and
slides 1-14 in Tutorial on varieties of Heyting algebras.)
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10 March 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Jonsson's Lemma, finitely generated varieties, finitely generated algebras, locally finite varieties, locally finite varieties have the FMP,
the Rieger-Nishimura lattice, (see Theorem 6.8 and Corollary 6.10 in Univ Alg. , Sections 2.3.5, 3.1.1, 3.1.2 , 4.1.2)
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10 March Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here
TUT 6.
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15 March 2015 Tuesday |
Lecture 9:00-11:00 SP B0.203 |
| Logics axiomatized by meet-implication formulas have the FMP, canonical varieties and Kripke completeness,
S4-algebras (also called closure algebras or interior algebras), the connection of closure algebras and Heyting algebras (Section 4.4 in
Notes on intuitionistic logic).
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17 March 2016 Thursday |
Lecture 13:00-15:00 SP B0.207 |
| Modal companions of intermediate logics (see slides 16-38 in
Crash course on intermediate logics and modal companions).
A short summary of the last 2-3 lectures can be found here Modal companions.
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17 March Feb 2016 Thursday |
Tutorial 15:00-17:00 SP B0.207 |
| The tutorial exercises can be found here
TUT 7.
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