Pasha Zusmanovich - teaching

Proof of the Dedekind theorem at the course of 2017/2018. Note the typo!

Pasha Zusmanovich

USPA1 Ordered Algebraic Structures, winter semester 2018/2019

A very brief synopsis:

Exam: Set 1

Set 2

Set 3

LITERATURE
Main:

S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition.

Additional:

C. Bergman, Universal Algebra, CRC Press, 2012.
G. Grätzer, The Congruences of a Finite Lattice. A Proof-by-Picture Approach, 2nd ed., Birkhäuser, 2016.
A.I. Mal'cev, Algebraic Systems, Springer, 1973.
Wikipedia: Boolean algebra, Distributive lattice, Lattice (order).

All the books are available in electronic form in multiple places.

= available in the university library

HOMEWORKS
Homeworks could be written in English or Czech, and submitted either in paper form or by email. If you write and not type, please be sure your handwriting is absolutely clear. Please do not forget to specify your name (if submitting on paper), the name of the course, and the number of the homework. All the points earned by doing homeworks (maximum 15-20 points for the whole semester) will be added to your score at the final exam.

Homework 1 (1 point)
Give an example of algebraic structure (different from those we had in the class!):

without proper substructures;
having exactly one substructure.

Compute the automorphism group of those structures.

Homework 2 (2 points)
1. What can be said about congruences of the algebraic structure \((\mathbb Z, N)\), where \(N\) is an unary operation given by \(N(n) = n+1\)?
2. Give an example of an algebraic structure (different from groups and rings) without nontrivial congruences.

Homework 3 (0.5 points)
Give an example of an ordered set which is not a lattice (different from those considered in the class).

Homework 4 (2 points)
Draw a Hasse diagram of the lattices of substructures of and congruences on the following lattices:
1. A totally ordered set of 3 elements.
2. \(N_5\).

Homework 5 (1 point)
Prove that any finite lattice contains 0 (the smallest element) and 1 (the greatest element).

Homework 6 (2 points)
Let G be a group. Is the lattice of all normal subgroups of G distributive? modular?

Homework 7 (1 point)
Prove that if \(a,b,c\) are elements of an arbitrary lattice such that \(a \le b\) and \(a \vee (b \wedge c) < b \wedge (a \vee c)\), then \((b \wedge (a \vee c)) \vee c = a \vee c\).
(This is a gap in the proof of the Dedekind theorem done in the class).

Homework 8 (1 point)
Prove that the lattice of divisors of a natural number n is complemented if and only if n is square-free.

Homework 9 (1 point)
Give an example of two algebraic systems \(A, B\) such that neither of them is isomorphic to a subsystem of their direct product \(A \times B\).

Homework 10 (1 point)
Prove that a linear order is directly indecomposable if and only if it consists of 1 element.

Homework 11 (3 points)

Finish to construct example of an (infinite) Boolean algebra not isomorphic to \(P(Y)\) for any \(Y\), started in the class: prove that for any infinite set \(X\), the Boolean algebra consisting of all finite and all cofinite subsets of \(X\), do not coincide with any \(P(Y)\).
Give another example of a Boolean algebra not isomorphic to \(P(Y)\) for any set \(Y\), different from those in the heading 1. (As one of the possible approaches you may consider a topological space \(X\), and try to construct a Boolean subalgebra of \(P(X)\) in terms of open/closed subsets).

Created: Sun Sep 24 2017
Last modified: Fri Jan 21 17:32:57 Central Europe Standard Time 2022