# Mathematical Structures in Logic

#### Universiteit van Amsterdam

The re-sit exam will take place on Monday, 26 June 10:00-13:15 at SP B0.204.

A handout on subvarieties of finitely generated varieties of Heyting algebras Handout 4. Apologies for ugly pictures!

Last year's take home exam.

## Practicalities

• Instructor: Nick Bezhanishvili, email: N.Bezhanishvili[at]uva.nl

• Teaching assistants: Mees de Vries, email: mees.devries[at]student.uva.nl and Frederik Lauridsen, email: f.m.lauridsen[at]uva.nl

• Time and Place: Lectures, Monday 10:00-11:45 (SP 904, A1. 04); Tutorials, Monday 12:00-12:45 (SP 904, A1. 04)

• EC: 8

• Assessment : There will be 7 Homework sheets and a final exam.

• Exam: There will be a final exam on 29 May, 10:00-13:00, (REC C1.04).

## Study materials

• A Course in Universal Algebra, Burris and Sankappanavar, 2012.

• Introduction to Lattices and Order, Second Edition, Davey and Priestley, Cambridge University Press, 2002.

• General Topology, J. Kelley, 2008.

• General Topology, S. Willard, 2004.

• Topology, J. Munkres, 2000.

• Stone Spaces, P. Johnstone, 1986.

• ## Homeworksheets

• Homework 1, due: 20 Feb before class.

• Homework 2, due: 6 March before class.

• Homework 3, due: 20 March before class.

• Homework 4, due: 3 April before class.

• Homework 5, due: 24 April before class.

• Homework 6, due: 8 May before class.

• Homework 7, due: 22 May before class.

• ## Lectures

 6 Feb 2017Monday Lecture10:00-12:00SP A1.04 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). 6 Feb 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 1-2. 13 Feb 2017Monday Lecture 10:00-12:00SP A1.04 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). 13 Feb 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 1-2. 20 Feb 2016Monday Lecture 10:00-12:00SP A1.04 Heyting algebras, equational definition, infinite distributive law, linear Heyting algebras, Heyting algebras of open sets of a topological space. (See Section 2.2.1 in here.) 20 Feb 2016Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 3-4. 27 Feb 2016Monday Lecture 10:00-12:00SP A1.04 Heyting algebras of upsets of a partially ordered set, Alexandroff topologies, interior algebras, topological insight on Goedel's embedding, Boolean algebra of regular open elements of a Heyting algebra, Glivenko's theorem. A short summary can be found here Handout 1 27 Feb 2016Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 3-4. 6 March 2017Monday Lecture 10:00-12:00SP A1.04 Elements of Universal algebra: HSP, varieties, Tarski's and Birkhoff's theorems, subdirectly irreducible algebras. A short summary can be found here Handout 2 6 March 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 5-6. 13 March 2017Monday Lecture 10:00-12:00SP A1.04 Connections between logics and varieties of algebras, Lindenbaum-Tarski algebras, quotient algebras, superintuionistic and intermediate logics, varieties of Heyting algebras, congruences, filters and ideals of Boolean algebras (Sections 11.11-11.16 and Sections 6.1-6.10 and 2.20-2.21 in Dav and Pries, for intermediate logics see Slides 1, see also the tutorial sheet 7-8 for the connection between logics and varieties of algebras). 13 March 2016Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 5-6. 20 March 2017Tuesday Lecture 10:00-12:00SP A1.04 Maximal, prime and ultrafilters of Boolean algebras, Prime filter theorem (Sections 10.7-10.15 in Lat and Ord). Note that in the lectures we worked mostly with filters, whereas Lat and Ord works mostly with ideals. 20 March 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 7-8. 27 March 2017Monday Lecture12:00-13:00SP A1.04 Stone representation theorem. (Sections 11.1-11.4 in Lat and Ord). Note again that in the lectures we worked with filters, whereas Lat and Ord works with ideals. 27 March 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 7-8. 3 April 2017Monday Lecture 10:00-12:00SP A1.04 Stone duality, Alexandroff and Stone-Cech compactifications of natural numbers (Sections 11.6 - 11.10 in Lat & Ord). Note that in Lat and Ord Stone spaces are called Boolean spaces. 3 April 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 9-10. 10 April 2017Monday Lecture 10:00-12:00SP A1.04 Priestley spaces, Priestley duality (Sections 11.7 - 11.27 in Lat & Ord). Note that in Lat and Ord Stone spaces are called Boolean spaces, also in the lectures we worked with filters, whereas Lat and Ord works with ideals. 10 April 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 9-10. 24 April 2017Monday Lecture 10:00-12:00SP A1.04 Priestley duality, Esakia spaces. (Sections 11.27 - 11.32 in Lat & Ord, see also the excellent notes of Pat Morandi on Duality in lattice theory, Sections 4-5.) 24 April 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 11-12. Please pay special attention to exercises 1 and 3. 1 May 2017Monday Lecture 10:00-12:00SP A1.04 Subdirectly irreducible HAs, Jonsson's Lemma (see Handout 1), the lattice of subvarieties of Var(A) for A a finite HA, locally finite varieties, The Rieger-Nishimura lattice, see Slides 2 1 May 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 11-12. Please pay special attention to exercises 1 and 3. 8 May 2017Monday Lecture 10:00-12:00SP A1.04 Finitely generated varieties (i.e., verities generated by one finite algebra, locally finite varieties, finitely approximable varieties (i.e. varieties that have the FMP), the finite model property of HA, some open problems on varieties of HAs, see Slides 3 8 May 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 13-14-15 15 May 2017Monday Lecture 10:00-12:00SP A1.04 Duality between modal algebras and modal spaces. Duality for K4 and S4-algebras. Connection between S4-algebras and Heyting algebras (see Sections 2.1-2.2 in Handout 3). 15 May 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 13-14-15 22 May 2017Monday Lecture 10:00-12:00SP A1.04 The Dedekind-MacNeille completions (Sections 7.36 - 7.44 in Lat and Ord.). We have not covered the completions of HAs. But if you are interested in this topic please consult Sections 1-2 in MacNeille Completions of HAs 22 May 2017Monday Tutorial 12:00-13:00SP A1.04 The tutorial exercises can be found here TUT 13-14-15