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The end depends upon the beginning...

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Title: The end depends upon the beginning...


1
  • The end depends upon the beginning...

2
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.

3
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
4
Centers
Center Cardinality
1. Beograd 27
2. Novi Sad 12
3. Niš 2
4. Priština 2
5
Decades
Years Cardinality
1. 60s 4
2. 70s 8
3. 80s 16
4. 90s 5
5. 2000s 10
Total 43
6
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
7
Top subject areas
Years Area
1. 70s Quasigroups
2. 80s Semigroups
3. 90s Semigroups
4. 2000s General algebra
8
Top centers
Years Centers
1. 70s Beograd
2. 80s Beograd
3. 90s Beograd
4. 2000s Novi Sad
9
Selected theses

10
Slaviša Prešic
  • A contribution to the theory of algebraic
    structures
  • Advisor Tadija Pejovic
  • Beograd, 1963.

11
Overview (Prešic)
  • 38 pages, Introduction 3 chapters
  • 18 bibliographic units

12
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)

13
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.

14
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.

15
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.

16
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.

17
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.

18
Svetozar Milic
  • A contribution to the theory of quasigroups
  • Advisor Slaviša Prešic
  • Beograd, 1971.

19
Overview (Milic)
  • 70 pages, Introduction 4 chapters
  • 52 bibliographic units

20
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.

21
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.

22
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.

23
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.

24
Dragica Krgovic
  • A contribution to the theory of regular
    semigroups
  • Advisor Mario Petrich
  • Beograd, 1982.

25
Overview
  • 70 pages, Introduction 4 chapters
  • 50 bibliographic units

26
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.

27
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.

28
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)

29
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.

30
  • That's all folks!
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