ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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Proof : The first thing to show is that the substructure is a semigroup. This follows from T62. Next we show that there is a neutral element. A substructure is per definition closed wrt each operation in the structure, and the neutral element is (considered to be ) an operation. The neutral element will therefore be a member of the substructure. A somewhat more operational way is 72. Theorem: Criterion for submonoid A subset B of en monoid (A, ♢ , e) is a submonoid if and kun if B is closed wrt the operation (B ♢ B ⊆ B) and e ∈ B. Even though the following result may seem somewhat meager it is useful to have it coined for later reference (T77). But logically it belongs right here: 73. Theorem: The trivial submonoid The singleton {e} is a submonoid Proof : It is obviously a substructure and therefore a submonoid. 74. Theorem: Structures induced by a monoid. If (A, ♢ , e) is a monoid and M is a set, then F(M, A) is a monoid with the induced operations. The neutral element is constant mapping M ∋ x ↦→ e. Product structures of monoids (Ai, ♢ i e = (e1, . . . , en). , ei) is a monoid. The neutral element is A quotient structure (A, ♢ , e) modulo ∼ of a monoid is a monoid. The neutral element is [e]∼. Proof : We only need to check the axioms: In all three cases we know that the underlying structure is a semigroup, which is the first axiom. Which remains is to check the neutral element. This is straightforward in all the cases. Lets do it. For F(M, A) the induced operation is the constant mapping defined on M with the constant value e. This is seen to be neutral by simple inspection.. 50
For product structures we must inspect e1 × . . . × en, which is the constant operation (e1, . . . , en), and is easily seen to be neutral : (a1, . . . , an) ♢ 1 ×· · ·× ♢ n (e1, . . . , en) = (a1 ♢ 1 e1, . . . , an ♢ n en) = (a1, . . . , an). For quotient structures we must inspect e/ ∼, which is the constant operation [e]. To show it to be neutral we have the following simple calculation: [a][e] = [ae] = [a] Examples of function spaces are X31,X32. Example quotient monoids are found in: E22 8. 3: Examples Deal with the following examples and exercises. E23 E24 E25 E26 X27 X28 X29 X30 X31 8. 4: Monoid homomorphisms 75. Definition: Monoid homomorphism A homomorphism between monoids is said to be a monoid homomorphism. X31,X33. X32 76. Definition: The kernel for a monoid homomorphism By the kernel for a monoid homomorphism we mean the inverse image of the neutral element. So if (A, ♢ , eA) and (B, ♡ , eB) are monoids and f : A → B is a monoid homomorphism, then f −1 ({eB}) is called the kernel for f (wrt to the structures). It is denoted by ker f 77. Theorem: The kernel of a monoid homomorphism is a submonoid Let (A, ♢ , eA) and (B, ♡ , eB) be monoids and f : A → B a monoid homomorphism, Then ker f is a submonoid. 51
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 and 48: is always possible to extend (doubl
Page 49: 67. Definition: Neutral element for
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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