ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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2 Induced semigroups 47 3 Powers 48 8 Monoids 1 Definitions 48 2 Induced monoids 49 3 Examples 51 4 Monoid homomorphisms 51 5 Inverse 52 6 Powers 52 7 Translations 53 9 Groups 1 Definitions 54 2 Induced group structures 55 3 subgroup 55 4 Quotient groups 59 5 Group homomorphisms 61 10 Rings 1 Definitions and rules 62 2 Subring 62 3 Induced rings 63 4 Quotient rings and ideals 64 5 Integral domains 66 11 Fields 1 Definitions 68 2 Quotient ring over a maximal ideal 69 3 Fields of fractions 69 12 Examples and exercises 13 Stikordsregister 4
1: Introduction The word algebra is of arabic origin meaning take into parts and put together, in short cut and paste. An equation is cut into pieces which are gathered to a new equation. And so algebra is about expressions put together using operations, originally the simple operations addition, subtraction, multiplication and division, and build together to create equations. Algebra is about manipulating such algebraic equations, with the intention to do it in a way which makes them solvable. The typical algebraic equation is an equation of the n-th degree in one unknown or a system of equations with several unknowns. The challenge to find methods for solution has been a main driving force in the development of mathematics. The specification of what is considered to be an algebraic problem has been widened in the course of this development. It has been recognized that a lot of specific methods can be subordinated as special cases of more general points of view. And today algebra is treated as an abstract discipline, where you do not necessarily define the objects which are the building stones or the operations used to connect them. In stead you make some assumptions of their properties and develop a theory which can be applied to all mathematical objects which have these properties. It is obvious that the more assumptions you make the more results may be deduced. On the other hand more assumptions will reduce the applicability. Therefore there exists a plethora of algebraic structures, and it is part of the sport to keep account of what precisely can be achieved with a specific choice of assumptions. So we start out with very few assumptions and gradually extend the number of assumptions. In this way you construct hierarchies of algebraic structures. Here we are going to focus on a single hierarchy, starting with the simplest possible configuration and then successively adding structure. This refinement of the structure can be accomplished by taking more operations into account and demanding more conditions on the operations. And now a survey over the text, which may be appropriate to bear in mind during reading, since it highlights the system behind the text. It might otherwise be a little more difficult to extract the main lines. The introductory chapters exhibit the fundamental notions for a lot of structures defined using operations in a set. There are the following categories of structures. 5
Page 1 and 2: Anders Madsen OPERATIONS AND STRUCT
Page 3: Table of contents 1 Introduction 2
Page 7 and 8: This is connected to the fact that
Page 9 and 10: Let f : B → A1 × . . . × An be
Page 11 and 12: Let ♢ and ♡ be operations on A
Page 13 and 14: to yield g ◦ ♡ = ♠ ◦ g, usi
Page 15 and 16: Let ♢ be an operation on A with n
Page 17 and 18: 2. 8: Induced operation on product
Page 19 and 20: Suppose that ♢ is an operation on
Page 21 and 22: 22. Definition: Compatibility Suppo
Page 23 and 24: 26. Theorem: Criterion for homomorp
Page 25 and 26: Proof : Assume f surjective and ♢
Page 27 and 28: Let (A, ♢ 1 , . . . , ♢ n ) and
Page 29 and 30: Proof : It follows from the definit
Page 31 and 32: Proof : The factors have per defini
Page 33 and 34: 23 SU(n) special unitary matrices 2
Page 35 and 36: 4. 3: Tables with homomorphisms In
Page 37 and 38: We consider the unary operation −
Page 39 and 40: Show that the determinant considere
Page 41 and 42: Lets take a look on the two example
Page 43 and 44: Anders Madsen CATEGORIES OF STRUCTU
Page 45 and 46: Use the table of contents in the fi
Page 47 and 48: is always possible to extend (doubl
Page 49 and 50: 67. Definition: Neutral element for
Page 51 and 52: For product structures we must insp
Page 53 and 54: Proof : The theorem is already prov
Page 55 and 56:
When dealing with abstract groups o
Page 57 and 58:
A homomorphism f : A ↦→ B is in
Page 59 and 60:
If H is a subgroup of the finite gr
Page 61 and 62:
Proof : This is just T91 111. Defin
Page 63 and 64:
10. 2: Subring 117. Definition: Del
Page 65 and 66:
A subset I is said to be et ideal o
Page 67 and 68:
133. Definition: Integral domain A
Page 69 and 70:
143. Theorem: Any substructure of a
Page 71 and 72:
Then it is routine to check that S
Page 73 and 74:
The prototype example of field exte
Page 75 and 76:
Consider the possibility of the map
Page 77 and 78:
Now it is easily checked that Fx is
Page 79 and 80:
We shall use this to illustrate an
Page 81 and 82:
What is said above may be generaliz
Page 83 and 84:
The field of fractions for the ring
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ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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