Universal Algebra: a reading course
This page is about the reading course `Universal Algebra' at the
University of Amsterdam (April - June 2006).
Contents of these pages
- The fifth and final homework set is available.
Deadline for submission: Monday July 10, evening.
- There will be no further meetings.
- Click here for the course material.
Material treated:
- June 27:
Algebra and Coalgebra Chapter:
sections 1,2,3,4,5,6(1,2).
- June 8: Chapter 4: 1-4, 6.
- May 29: Chapter 2: 10,11, 14.
- May 8: Chapter 2: 5-9.
- April 24: Chapter 1: 1-5, Chapter 2: 1-3.
- Fifth homework set: (All numbers refer to
this version of the Algebra/Coalgebra Chapter).
(Deadline for submission: Monday July 10, evening.)
- Prove Proposition 1.30.
- Let for any frame S, f be the map sending a state s to the
principal ultrafilter generated by s.
Prove that f is a bounded morphism iff S is image finite.
- (a)Show that the algebra of Example 1.69 is not simple.
(b)Give a direct algebraic proof that the algebra of Example 1.70
is not s.i.
- Fill in the details of Theorem 1.85.
- Fourth homework set:
Exercises from Chapter IV:
(1.4 or 1.9) and 2.7 and (3.1 or 3.3) and 4.7 and 6.4.
Deadline for submission: Monday June 19, noon.
- Third homework set:
- Prove from first principles (i.e. only use results proved in
section 10) that if F is free for K over X, then
F is also free for HSP(K) over X.
- Let M be the variety of monadic (or S5-)algebras; you may
think of M as the class of subalgebras of complex
algebras of frames (W,R) in which R is an equivalence relation.
(a) Prove that M is locally finite.
(b) (BONUS) Make exercise 11.3.
- Make exercise 11.4
- Make exercise 14.2.
- Make either exercise 12.4 or exercise 14.12, at your choice.
If you choose the latter one, write your derivation so that each
equation is numbered, and explicitly justified as either an
axiom or the application of a derivation
rule to earlier equations.
Deadline for submission: Thursday June 1, noon.
- Second homework set:
Exercises II.5.5, II.6.6, II.7.3, II.8.3, II.9.2.
Deadline for submission: Thursday May 18.
- First homework set: Exercises I.1.10, I.4.6, I.5.8, II.1.1.
Deadline for submission: Tuesday May 2 at the beginning of the modal
logic class.
Organization
- There will be no classes as such.
- Every two/three weeks a part of the book is assigned for reading.
- Every two/three weeks there is a meeting in which students can ask
questions concerning this material.
- Preferrably students should notify the teacher of their
questions at least one day before the meeting.
- Around the time of the meeting, homework assignments will be
made available. Students will have two weeks time to complete
the exercises.
Lecturer
Dates/location
- In principle, meetings will be held on Wednesdays, at 3.15 pm.
Grading
- Students can obtain six credit points (EC) for the course.
- The grading will be through homework assignments.
One of the aims of universal algebra is to extract the common
elements of seemingly different types of algebraic structures
such as groups, rings or lattices. Doing so one discovers general
concepts, constructions, and results which unify and generalize
the known special situations. Applications of universal algebra
can be found in logic through the interface of algebraic logic.
The course will introduce the students to the basic concepts and
theory of universal algebra.
-
We will primarily make use of the book A Course in Universal Algebra
by Burris and Sankappanavar.
This book is out of sale now, but an updated, electronic version is
available
here.
- Tentatiely, material covered during the course will consist of
- Chapter 1,
- Chapter 2: sections 1-3,4-11, 14,
- Chapter 4: 1-4,6,9,
- Chapter 5: 1-2.
Yde Venema