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- 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. Ðuro Kurepa 6

2. Slaviša Prešic 5

3. Svetozar Milic 5

4. Žarko Mijajlovic 3

5. Branka Alimpic 3

5. Siniša Crvenkovic 2

6. Miroslav Ciric 2

7. Dragan Mašulovic 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 Prešic

- A contribution to the theory of algebraic

structures - Advisor Tadija Pejovic
- Beograd, 1963.

Overview (Prešic)

- 38 pages, Introduction 3 chapters
- 18 bibliographic units

Chapter 2 (Prešic)

- 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 - (xy)zx(yz)

Chapter 2 (Prešic)

- This can be seen as follows. If xya and yzb,
- Then azxb.
- If, instead of xya 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 (Prešic)

- 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 (Prešic)

- 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))---gt(a,c)). Some examples are also

given.

Chapter 3 (Prešic)

- In this chapter some estimates of the number of

different algebras of the given type, satisfying

algebraic laws of the form wu 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 ... pk1)gtB(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 (Prešic)

- 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 ngt1 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 Milic

- A contribution to the theory of quasigroups
- Advisor Slaviša Prešic
- Beograd, 1971.

Overview (Milic)

- 70 pages, Introduction 4 chapters
- 52 bibliographic units

Chapter 1 (Milic)

- 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 (Milic)

- 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 (Milic)

- This chapter is devoted to the discussion of the

generalized (i,j)-modular systems of

n-quasigroups. For example the general solution

of the functional equation - A(x1,...,xi-1,B(y1,...,yn),xi1,...,xn)
- C(y1,...,yj-1,D(x1,...,xi-1,yj,xi1,...,xn),yj1,

...,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 kind-they all come from some Abelian

group.

Chapter 4 (Milic)

- 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 Krgovic

- A contribution to the theory of regular

semigroups - Advisor Mario Petrich
- Beograd, 1982.

Overview

- 70 pages, Introduction 4 chapters
- 50 bibliographic units

Chapter 1 (Krgovic)

- 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 aaxa - it is (m,n)-regular if for every a in S there

exists an x in S such that aamxan.

Chapter 2 (Krgovic)

- 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 (Krgovic)

- 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 (Krgovic)

- 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!