ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

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An operation ♢ in A with 0 operands is uniquely determined by ♢ (()) that is a certain element of A, which the operation can be identified with. The operations with 0 operands are therefore sometimes called the constant operations. So we shall all the time think of an operation with 0 operands as an element of A. 3. Remark: Operation symbols To underline the general aspect we are going to use operation symbols which do not already have a definite meaning such as x ♢ y, x ♡ y. 4. Definition: Prefix, Infix, Postfix We have a number of alternative ways to denote an operation, other than ♢ (a1, . . . , an), namely ♢ a1 . . . an, a1 ♢ . . . ♢ an, a1 . . . an ♢ called respectively prefix, infix and postfix notation. In some contexts we do not state which notation we use in the hope that it is otherwise easily understood from the context. For unary operations only prefix and postfix notation are relevant. Generally we use the operator notation. Infix notation is mostly used for binary operations. 5. Remark: Backwards polish Postfix notation is in principle the simplest and is used in programming contexts, for instance on some pocket calculators. It is simply because you can avoid parentheses E1 X2 2. 4: Homomorphisms We know from the real numbers the the rule exp(x + y) = exp(x) · exp(y). For a linear mapping L we have that L(x + y) = L(x) + L(y). If we let M(X) denote the matrix which belongs to a certain rotation X, then M(X ◦ Y ) = M(X) · M(X). In each of these three cases we are dealing with a mapping with a particularly nice behaviour with respect to a couple of operations. This may be generalized to arbitrary pairs of operations if they have the same arity: 6. Definition: Homomorphism wrt a pair of operations 10
Let ♢ and ♡ be operations on A and B respectively and both having n operands. Let f : A → B be a mapping, so that for all (a1, . . . , an) ∈ A n : you have that f( ♢ (a1 . . . an)) = ♡ (f(a1) . . . f(an)) Then we say that f is homomorphic wrt ( ♢ , ♡ ). 7. Remark: Homomorphisms wrt unary and 0-ary In infix notation the defining equation may be written f(a1 ♢ . . . ♢ an) = f(a1) ♡ . . . ♡ f(an) which in the binary case is f(a1 ♢ a2) = f(a1) ♡ f(a2) In the unary case it means that f( ♢ a) = ♡ f(a), and for n = 0 we get f( ♢ ) = ♡ Loosely we can say that it doesn’t matter if we carry out the operation before using the mapping or first map all the operands and then use the operation on the result. This can be most elegantly and concentrated stated by the equation ⋆ f ◦ ♢ = ♡ ◦ f which is displayed in the diagram ♢ A n f ✲ B n ♡ ❄ ❄ f A ✲ B 8. Definition: Isomorphism wrt a pair of operations Let ♢ and ♡ be operations in A and B both with n operands. Let f : A → B be a bijective mapping such that f is a homomorphism wrt ( ♢ , ♡ ) and f −1 is a homomorphism wrt ( ♡ , ♢ ) Then we say that f is an isomorphism wrt ( ♢ , ♡ ). 11
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: Let f : B → A1 × . . . × An be
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
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Proof : This is just T91 111. Defin
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10. 2: Subring 117. Definition: Del
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A subset I is said to be et ideal o
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133. Definition: Integral domain A
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143. Theorem: Any substructure of a
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Then it is routine to check that S
Page 73 and 74:
The prototype example of field exte
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Consider the possibility of the map
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Now it is easily checked that Fx is
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We shall use this to illustrate an
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What is said above may be generaliz
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The field of fractions for the ring
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