1 / 22

Chapter 6 Abstract algebra

Chapter 6 Abstract algebra. Groups  Rings  Field  L attics and Boolean algebra. 6.1 Operations on the set

barrettr
Télécharger la présentation

Chapter 6 Abstract algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6 Abstract algebra • Groups • Rings • Field • Lattics and Boolean algebra

  2. 6.1 Operations on the set • Definition 1：An unary operation on a nonempty set S is an everywhere function f from S into S; A binary operation on a nonempty set S is an everywhere function f from S×S into S; A n-ary operation on a nonempty set S is an everywhere function f from Sn into S. • closed

  3. Associative law: Let be a binary operation on a set S. a(bc)=(ab)c for a,b,cS • Commutative law: Let be a binary operation on a set S. ab=ba for a,bS • Identity element: Let be a binary operation on a set S. An element e of S is an identity element if ae=ea=a for all a S. • Theorem 6.1: If  has an identity element, then it is unique.

  4. Inverse element: Let  be a binary operation on a set S with identity element e. Let a S. Then b is an inverse of a if ab = ba = e. • Theorem 6.2: Let  be a binary operation on a set A with identity element e. If the operation is Associative, then inverse element of a is unique when a has its inverse

  5. Distributive laws: Let  and  be two binary operations on nonempty S. For a,b,cS, • a(bc)=(ab)(ac), (bc)a=(ba)(ca)

  6. Definition 2: An algebraic system is a nonempty set S in which at least one or more operations Q1,…,Qk(k1), are defined. We denoted by [S;Q1,…,Qk]. • [Z;+] • [Z;+,*] • [N;-] is not an algebraic system

  7. Definition 3: Let [S;*] and [T;] are two algebraic system with a binary operation. A function  from S to T is called a homomorphism from [S;*] to [T;] if (a*b)=(a)(b) for a,bS.

  8. Theorem 6.3 Let  be a homomorphism from [S;*] to [T;]. If  is onto, then the following results hold. • (1)If * is Associative on S, then  is also Associative on T. • (2)If * is commutative on S, then  is also commutation on T • (3)If there exist identity element e in [S;*],then (e) isidentity element of [T;] • (4) Let e be identity element of [S;*]. If there is the inverse element a-1 of aS, then (a-1) is the inverse element (a).

  9. Definition 4: Let  be a homomorphism from [S;*] to [T;].  is called an isomorphism if  is also one-to-one correspondence. We say that two algebraic systems [S;*] and [T;] are isomorphism, if there exists an isomorphic function. We denoted by [S;*][T;](ST)

  10. 6.2 Semigroups,monoids and groups • 6.2.1 Semigroups, monoids • Definition 5: A semigroup [S;] is a nonempty set together with a binary operation  satisfying associative law. • Definition 6: A monoid is a semigroup [S; ] that has an identity.

  11. Let P be the set of all nonnegative real numbers. Define & on P by • a&b=(a+b)/(1+ab) • Prove［P;&］is a monoid.

  12. 6.2.2 Groups • Definition 7: A group [S; ] is a monoid, and there exists inverse element for aS. (1)for a,b,cS,a (b c)=(a b) c; (2)eS,for aS,a e=e a=a; (3)for aS, a-1S, a a-1=a-1 a=e

  13. [R-{0},] is a group • [R,] is a monoid, but is not a group • [R-{0},], for a,bR-{0},ab=ba • Abelian (or commutative) group • Definition 8: We say that a group [G;]is an Abelian (or commutative) group if ab=ba for a,bG. • [R-{0},],[Z;+],[R;+],[C;+]are Abelian (or commutative) group . • Example: Let [G; ] be a group with identity e. If xx=e for xG, then [G; ] is an Abelian group.

  14. Example: Let G={1,-1,i,-i}.

  15. G={1,-1,i,-i}, finite group • [R-{0},],[Z;+],[R;+],[C;+]，infinite group • |G|=n is called an order of the group G • Let G ={ (x; y)| x,yR with x 0} , and consider the binary operation ● introduced by (x, y) ● (z,w) = (xz, xw + y) for (x, y), (z, w) G. • Prove that (G; ●) is a group. • Is (G;●) an Abelian group?

  16. [R-{0},] , [R;+] • a+b+c+d+e+f+…=(a+b)+c+d+(e+f)+…, • abcdef…=(ab)cd(ef)…, • Theorem 6.4: If a1,…,an(n3), are arbitrary elements of a semigroup, then all products of the elements a1,…,an that can be formed by inserting meaningful parentheses arbitrarily are equal.

  17. a1  a2 …  an If ai=aj=a(i,j=1,…,n), then a1 a2 … an=an。 na

  18. Theorem 6.5: Let [G;] be a group and let aiG(i=1…,n). Then • (a1…an)-1=an-1…a1-1

  19. Theorem 6.6: Let [G;] be a group and let a and b be elements of G. Then • (1)ac=bc, implies that a=b(right cancellation property)。 • (2)ca=cb, implies that a=b。(left cancellation property) • S={a1,…,an}, al*aial*aj(ij)， • Thus there can be no repeats in any row or column

  20. Theorem 6.7: Let [G;] be a group and let a,b, and c be elements of G. Then • (1)The equation ax=b has an unique solution in G. • (2)The equation ya=b has an unique solution in G.

  21. Let [G;] be a group. We define a0=e， • a-k=(a-1)k，ak=a*ak-1(k≥1) • Theorem 6.8: Let [G;] be a group and a G, m,n Z. Then (1)am*an=am+n (2)(am)n=amn • a+a+…+a=ma, ma+na=(m+n)a n(ma)=(nm)a

  22. Next: Permutation groups • Exercise P348 (Sixth) ORp333(Fifth) 9,10,11,18,19,22,23, 24; • P355 (Sixth) ORP340(Fifth) 5—7,13,14,19—22 • P371 (Sixth) ORP357(Fifth) 1,2,6-9,15, 20 • Prove Theorem 6.3 (2)(4)

More Related