ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

More documents

Recommendations

Info

4. 2: Tables with operations . Operations with 0 operands Set operation Closed subsets 1 N 0 kN 2 C 0 R, Q, Z 3 C 1 R, Q, Z, N 4 C i 5 Mn(L) En, E SLn(L), On(L), Pern(L) 6 Mn(C) En, E Mn(R), Mn(Q), Mn(Z) 7 D(X) ∅ 8 F(X, X) I 9 ⟨a, b, c, . . .⟩ (), empty word Unary operations Set operand operation comment closed subsets 1 C a −a Opposite R, Q, Z, N 2 C ⋆ a a −1 Reciprocal 3 C a ā conjugate 4 L n x −x Opposite 5 Mn(L) A A † ,A t ,A ⊤ transposed 6 Mn(C) A A ⋆ adjoint 7 F(X, X) f f −1 inverse 8 D(X) A X \ A complement Binary operations Set operands operation name closed subsets 1 C a, b a + b R, Q, Z, N 2 C ⋆ a, b ab 3 Mn(L) A, B A + B 4 Sn s, t st 5 D(X) A,B A ∩ B 6 D(X) A,B A ∪ B 34
4. 3: Tables with homomorphisms In the following tables A, X, S, M, G, R, L, V are denoting an arbitrary structure of the following types respectively general structure, set, semigroup, monoid, group, ring, field. If we have more structures of the same type we shall use indexed letters in the usual way, so that M1, M2, . . . etc denote sets. We let M ′ denote a subset of M and so on. The set of homomorphisms of a structure A into a structure B is denoted by Hom(A, B). Homomorphisms wrt binary operations. x ♢ y A x f(x) B u ♡ v symbo 1 x + y Rn x Ax Rm u + v 2 m + n Z n an R uv exp. m 3 A + B Mat(m, n, L) A L(A, V, W) Lin(V, W ) u + v ass. lin. m 4 L + M Lin(V, W ) L M(L, V, W) Mat(m, n, L) u + v ass. ma 5 LM Lin(V, W ) L M(L, V, W) Mat(m, n, L) u ◦ v ass. ma 6 P + Q L[X] P P (L) Lin(V, V ) u + v 7 x + y Rn x Ax Rm u + v mult. m. 8 xy N x σ(x) N0 N u + v prime spec 9 xy OrdA x |x| N u + v word len 10 x × y Finite sets x |x| N0 uv ant. ele 11 x ◦ y Rot(3) x ˆx Mb u ◦ v ass. Möbius tran 12 x ◦ y Mb x ˆx Rot(3) u ◦ v ass. ro 13 xy SU(2) x ˆx SO(3) uv ass. ro 14 5: Examples and exercises Example 1: Different ways of notation. (see nr 5) We have that +(a, ·(b, c)) = +a(·bc) = a + b · c = abc · + Exercise 2: Backward polish. (see nr 5) 10 2 · 5 + √ 2 · 10 : = 1 Example 3: Linear mappings are homomorphisms (see nr 8) 35
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: 23 SU(n) special unitary matrices 2
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

ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?