ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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You should realize that you have already met a number of homomorphisms by studying the following examples and exercises: E3 E4 E5 E6 E7 E8 X9 X10 We could say that a homomorphism preserves the algebraic structure and it should then not be surprising that a composition of homomorphisms is again a homomorphism. There are other rules for combining homomorphisms. These are the subject of the next section: 9. Theorem: Composition and homomorphisms Let ♢ , ♡ and ♠ be operations on A,B and C respectively, all with n operands, And let f : A → B, g : B → C be mappings with h = g ◦f, which we summarize in the diagram Then A ✒ f B ❅ g ❅❅❘ h ✲ C 1) If f and g are homomorphisms then so is h. 2) If f and h are homomorphisms and f is surjective then g is a homomorphism. 3) If g and h are homomorphisms and g is injective then f is a homomorphism. Proof : We use the compact homomorphism equation ⋆ in D6. Then the proof of (1) takes the form h ◦ ♢ = g ◦ f ◦ ♢ = g ◦ ♡ ◦ f = ♠ ◦ g ◦ f = ♠ ◦ g ◦ f = ♠ ◦ h The proof of (2) consists in combining the two equations g ◦ ♡ ◦ f = g ◦ f ♢ = h ◦ ♢ = ♠ ◦ h, ♠ ◦ g ◦ f = ♠ ◦ h 12
to yield g ◦ ♡ = ♠ ◦ g, using that f and so f is surjective. Regarding (3) we combine the equations g ◦ f ◦ ♢ = h ◦ ♢ g ◦ ♡ ◦ f = g ◦ f ◦ ♢ = h ◦ ♢ to deduce that f ◦ ♢ = ♡ ◦ f, since g is injective. Since these proofs may seem very formalistic and concentrated we repeat them for the binary case in a more simple form Proof of (1): Let a1, a2 ∈ A then h(a1 ♢ a2) =g(f(a1 ♢ a2)) Proof of (2): Let b1, b2 ∈ B. =g(f(a1) ♡ f(a2)) = g(f(a1)) ♠ g(f(a2)) =h(a1) ♠ h(a2) As f is surjective we can choose a1, a2 ∈ A such that f(a1) = b1, f(a2) = b2. We then have that g(b1 ♡ b2) =g(f(a1) ♡ f(a2)) = g(f(a1 ♢ a2)) = h(a1 ♢ a2) Proof of (3): =h(a1) ♠ h(a2) = g(f(a1)) ♠ g(f(a2)) =g(b1) ♠ g(b2) Let a1, a2 ∈ A. First we see that g(f(a1 ♢ a2)) = h(a1 ♢ a2) = h(a1) ♠ h(a2) and next that g(f(a1) ♡ f(a2)) = g(f(a1)) ♠ g(f(a2)) = h(a1) ♠ h(a2) 13
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: Let ♢ and ♡ be operations on A
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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