Ordered algebraic structures

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Lecturers:

Prof. Dr. Jakob Cimprič, Assoc. Prof. Dr. Karin Cvetko Vah

Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Partially ordered sets. Modular lattices. Distributive lattices and their representations. Boolean algebras and their representations.
Partially ordered groups and vector spaces. Convex subgroups. Homomorphisms. Archimedean and Dedekind complete groups. Linearly ordered groups.Partially ordered rings. Orderings on the field of fractions. Formally real fields. Real closed fields. Archimedean orderings. Orderings and valuations.

Readings

G. Birkhoff: Lattice Theory, 3rd edition, AMS, Providence, 2006.
T.S. Blyth: Lattices and Ordered Algebraic Structures, Springer, 2005.
L. Fuchs: Partially Ordered Algebraic Systems, Pergamon Press, London, 1963.
A. M. W. Glass: Partially Ordered Groups, World Scientific, River Edge, 1999.
B. Lavrič: Delno urejene grupe in delno urejeni kolobarji, DMFA-založništvo, Ljubljana, 1993.
B. Lavrič: Delno urejeni vektorski prostori, DMFA-založništvo, Ljubljana, 1995.

Objectives and competences

The student learns the basics of the theory of ordered algebraic structures.

Intended learning outcomes

Knowledge and understanding:
Understanding of basic concepts and theorems of the theory of ordered algebraic structures, and their role in some other areas.
Application:
In other mathematical areas.
Reflection:
Understanding the theory on the basis of examples and applications.
Transferable skills:
Formulation and solution of problems using abstract methods.

Learning and teaching methods

Lectures, exercises, homeworks, consultations.

Assessment

Homeworks
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Jaka Cimprič:
CIMPRIČ, Jaka. Free skew fields have many [ast]-orderings. Journal of algebra, ISSN 0021-8693, 2004, vol. 280, no. 1, str. 20-28. [COBISS-SI-ID 13210201]
CIMPRIČ, Jaka, KLEP, Igor. Generalized orderings and rings of fractions. Algebra universalis, ISSN 0002-5240, 2006, vol. 55, no. 1, str. 93-109. [COBISS-SI-ID 13966937]
CIMPRIČ, Jaka. A representation theorem for archimedean quadratic modules on [star]-rings. Canadian mathematical bulletin, ISSN 0008-4395, 2009, vol. 52, št. 1, str. 39-52. [COBISS-SI-ID 15084633]
Karin Cvetko Vah:
CVETKO-VAH, Karin. Internal decompositions of skew lattices. Communications in algebra, ISSN 0092-7872, 2007, vol. 35, no. 1, str. 243-247. [COBISS-SI-ID 14223193]
CVETKO-VAH, Karin. On strongly symmetric skew lattices. Algebra universalis, ISSN 0002-5240, 2011, vol. 66, no. 1-2, str. 99-113. [COBISS-SI-ID 16219993]
BAUER, Andrej, CVETKO-VAH, Karin. Stone duality for skew Boolean algebras with intersections. Houston journal of mathematics, ISSN 0362-1588, 2013, vol. 39, no. 1, str. 73-109. [COBISS-SI-ID 16620377]