Transcript Document

The end depends upon the beginning...
An overview of history of algebra in
Serbia
This overview is based on the analysis of Ph. D.
theses in the field of algebra defended in Serbia.
Most of these theses are available at the Virtual
library (Faculty of Mathematics, Belgrade).
Mathematics genealogy project was also used to
acquire additional data. Only theses defended in
Serbia were taken into account.
Areas
Subject area
Cardinality
1. Quasigroups
6
2. Semigroups
13
3. Groups
2
4. Rings
2
5. Fields
1
6. Numbers
4
7. Algebraic equations and
algebraic geometry
5
8. General algebra
10
Total
43
Centers
Center
Cardinality
1.
Beograd
27
2.
Novi Sad
12
3.
Niš
2
4.
Priština
2
Decades
Years
Cardinality
1.
60s
4
2.
70s
8
3.
80s
16
4.
90s
5
5.
2000s
10
Total
43
Top advisors
Advisor
Cardinality
1.
Đuro Kurepa
6
2.
Slaviša Prešić
5
3.
Svetozar Milić
5
4.
Žarko Mijajlović
3
5.
Branka Alimpić
3
5.
Siniša Crvenković
2
6.
Miroslav Ćirić
2
7.
Dragan Mašulović
2
Top subject areas
Years
Area
1.
70s
Quasigroups
2.
80s
Semigroups
3.
90s
Semigroups
4.
2000s
General algebra
Top centers
Years
Centers
1.
70s
Beograd
2.
80s
Beograd
3.
90s
Beograd
4.
2000s
Novi Sad
Selected theses
Slaviša Prešić
A contribution to the theory of algebraic structures
Advisor: Tadija Pejović
Beograd, 1963.
Overview (Prešić)
38 pages, Introduction + 3 chapters
18 bibliographic units
Chapter 2 (Prešić)
Suppose we have an algebraic structure with
some operations. Let us concentrate on the
simple example of a semigroup.
So we have a set with one binary operation which
satisfy the following law:
(x*y)*z=x*(y*z)
Chapter 2 (Prešić)
This can be seen as follows. If: x*y=a and y*z=b,
Then a*z=x*b.
If, instead of x*y=a we write r(x,y,a) we can write
the condition for associativity as:
if r(x,y,a) and r(y,z,b) and r(a,z,c) then r(x,b,c)
if r(x,y,a) and r(y,z,b) and r(x,b,c) then r(a,z,c)
If we forget about the origin of our definition of
r(x,y,a), we arrive at the notion of an associative
relation.
Chapter 2 (Prešić)
Suppose we have an arbitrary ternary relation.
Can we find the smallest associative relation
which contains this one? The chapter 2 is devoted
to showing that the answer is yes, but not only for
this simple case, but for more general cases of
arbitrary relations (not necessarily ternary
relations) arising from various algebraic laws so
satisfying quite general laws.
Chapter 2 (Prešić)
The idea is to define a partial (ternary or what is
appropriate for the case upon investigation)
operation which is defined on the set of relations
and by iteration we arrive at the solution. This idea
is a generalization of the idea of finding smallest
transitive relation which contains the given one (in
this case, the operation at the foundation of this
proof is ((a,b),(b,c))--->(a,c)). Some examples are
also given.
Chapter 3 (Prešić)
In this chapter some estimates of the number of
different algebras of the given type, satisfying
algebraic laws of the form w=u such that the same
letters should appear in w and u, on a set of n
elements are given. If we denote that number by
B(n), then the inequality which is the fundamental
one, and from which the others are derived is the
following:
B(p1+ ... +pk+1)>B(p1) . . . B(pk),
where pi are different numbers. The most general
result depends on the number of different
presentation of a given number as a sum of
natural numbers.
Chapter 4 (Prešić)
This chapter is devoted to the study of the relation
between an algebra and its group of
automorphisms. The main theorem in this chapter
is the following.
If G is an arbitrary group and n>1 then one can
define an n-ary operation f on this group such that
the group of automorphisms of (G,f) is exactly the
group G.
Svetozar Milić
A contribution to the theory of quasigroups
Advisor: Slaviša Prešić
Beograd, 1971.
Overview (Milić)
70 pages, Introduction + 4 chapters
52 bibliographic units
Chapter 1 (Milić)
Chapter 1 is mostly reserved for necessary
notation and the recollection of known results
concerning quasigroups which are needed in the
subsequent chapters.
Chapter 2 (Milić)
In Chapter 2 various systems of quasigroups
satisfying various algebraic laws of special types
have been investigated. For some of these cases
it has been proved that these quasigroups are
isotopic to some group. Method of the proof may
be used for laws not necessarily of associative
type.
Chapter 3 (Milić)
This chapter is devoted to the discussion of the
generalized (i,j)-modular systems of nquasigroups. For example the general solution of
the functional equation
A(x1,...,xi-1,B(y1,...,yn),xi+1,...,xn)=
=C(y1,...,yj-1,D(x1,...,xi-1,yj,xi+1,...,xn),yj+1,...,yn)
on n-quasigroups is given. In addition to that, it
has been proved that the quasigroups satisfying
all (i,j)-modular laws are of the very simple kindthey all come from some Abelian group.
Chapter 4 (Milić)
In this chapter the main interest lies in the
investigation of generalized groupoids with
division which satisfy balanced algebraic law.
Some of the results which have been proved in
this chapter are convenient for application for
solving functional equation of general
associativity. Some examples were also
presented.
Dragica Krgović
A contribution to the theory of regular semigroups
Advisor: Mario Petrich
Beograd, 1982.
Overview
70 pages, Introduction + 4 chapters
50 bibliographic units
Chapter 1 (Krgović)
Some definitions and known results are listed
Semigroup S is regular if for every a in S there
exists an x in S such that a=axa;
it is (m,n)-regular if for every a in S there exists an
x in S such that a=amxan.
Chapter 2 (Krgović)
Some characterizations of regular and
(m,n)-regular semigroups are given. They
generalize previously known results. Maybe the
most interesting are the results that characterize
when the given regular semigroup is a union of
groups.
Chapter 3 (Krgović)
Completely 0-simple semigroups are
characterized using 0-minimal bi-ideals
(semigroup with 0 is completely simple if the
product is not trivial, it does not contain any ideal
except the zero ideal, and it contains a primitive
idempotent)
Chapter 4 (Krgović)
In this chapter, the problem of bi-ideal extension is
discussed: Given a semigroup S and a semigroup
with 0 Q, is there a semigroup V which contains a
bi-ideal S' isomorphic to S, such that V/S' is
isomorphic to Q.
That's all folks!