The end depends upon the beginning...

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• The end depends upon the beginning...

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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.

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

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Slaviša Prešic

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

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Overview (Prešic)

• 38 pages, Introduction 3 chapters
• 18 bibliographic units

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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)

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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.

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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.

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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.

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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.

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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.

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Svetozar Milic

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

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Overview (Milic)

• 70 pages, Introduction 4 chapters
• 52 bibliographic units

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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.

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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.

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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.

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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.

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Dragica Krgovic

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

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Overview

• 70 pages, Introduction 4 chapters
• 50 bibliographic units

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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.

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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.

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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)

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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.

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• That's all folks!