The end depends upon the beginning An overview

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 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. 60 s 4 2. 70 s 8 3. 80 s 16 4. 90 s 5 5. 2000 s 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. 70 s Quasigroups 2. 80 s Semigroups 3. 90 s Semigroups 4. 2000 s General algebra

Top centers Years Centers 1. 70 s Beograd 2. 80 s Beograd 3. 90 s Beograd 4. 2000 s Novi Sad

Selected theses

Slaviša Prešić A contribution to 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(p 1+. . . +pk+1)>B(p 1). . . 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 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(x 1, . . . , xi-1, B(y 1, . . . , yn), xi+1, . . . , xn)= =C(y 1, . . . , yj-1, D(x 1, . . . , 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 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!