ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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Proof : Associativity is inherited by the induced structures, and no more is needed. 7. 3: Powers The notion of power with positive integer exponent can be defined and the power rules proved at this very primitive level. And then we have dealt with this aspect once and for always. Notice that the definitions and the proofs are the ones you learned very early in your life. 65. Definition: Powers with positive exponent. In a semigroup parentheses are superfluous and so powers with positive integer exponent of an element a may be immediately defined by the following recursive procedure: a 1 = a, a n+1 = aa n 66. Theorem: Rules for calculating with Powers a n a m = a n+m , (a n ) m = a mn for all a and all n, m > 0. If ab = ba then (ab) n = a n b n . Proof : Induction. Example. Let m be fixed. We show the formula for this m and all n by induction after n. For n = 1 we have u n u m = uu m = u m+1 per definition of power. Suppose then that you have proved the formula for n = k. Then we have that u k+1 u m = uu k u m = uu m+k = u m+k+1 , and so the formula is also proved for n = k + 1. 8: Monoids 8. 1: Definitions A very large part of the binary operations, used in algebraic structures, have a neutral element, an element without action when used by the operation. To study the pure effect of introducing a neutral element we introduce the notion of a monoid, a structure which differs from a semigroup only by admitting a neutral element. Actually there are a lot of interesting monoids without further structure. For the sake of completeness we take the following 48
67. Definition: Neutral element for a binary operation The element e is said to be a neutral element wrt den binary operation ♢ on A, if for all a ∈ A we have that a ♢ e = e ♢ a = a. We identify e with the constant operation (with 0 operands) for which e is the constant value. The prototype of a ”genuine” monoid, one without further ado is the set F(X, X) of all mappings of X into itself with composition as binary operation and the identity mapping as the neutral element. Other examples are seen in X33 68. Theorem: Uniqueness of neutral element Any operation has at most one neutral element. Proof : If e and e ′ are neutral elements, then e = e ♢ e ′ = e ′ 69. Definition: Monoid A monoid is an algebraic structure (A, ♢ , e) for which (A, ♢ ) is a semigroup (the underlying semigroup) and e is a neutral element wrt ♢ . Since the neutral element is uniquely determined we shall also let (A, ♢ ) denote the monoid. 8. 2: Induced monoids 70. Definition: Submonoid A substructure of a monoid which is itself a monoid with the induced structure, is said to be a submonoid. 71. Theorem: Each substructure of a monoid is a submonoid Substructure of monoid is submonoid 49
Page 1 and 2: Anders Madsen OPERATIONS AND STRUCT
Page 3 and 4: Table of contents 1 Introduction 2
Page 5 and 6: 1: Introduction The word algebra is
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: is always possible to extend (doubl
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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