The basic properties of an algebraic system
basic requirements:
- Composite function definition - the definition of the operation to meet the unique mapping
- Closed
1 relates to a binary operation
Commutative law
Associativity left and the right combination of binding proof does not satisfy the associative law only to find counter-examples
Idempotent law
Erasing law
2 relates to two different binary operations
Distributive property (binary operation), the left and the right dispensing assigned
eg <X, *, +>, * + to meet the distributive property, then there should be:
Any x1, x2, x3 belonging to X,
Left assignment: x1 * (x2 + x3) = x1 * x2 + x1 * x3
&&
Right distribution: (x2 + x3) * x1 = x2 * x1 + x3 * x1
or prove that only one side, again proved the establishment of exchange law
Absorption law - for two commutative operations
Specific elements of the binary operation 3
Yuan Yuan && unitary unitary left and right identity element
Zero dollars left and right zero dollars zero dollars &&
Idempotents operation table as an element corresponding to the diagonal elements called columns Idempotents
Yuan Yuan and its inverse reversible
Second, isomorphism & homomorphism
- Solving isomorphism / homomorphism between two algebraic system
- Determining whether a function is isomorphic (same state) between the two algebraic system
- F is a homomorphism proved の step V1 to V2:
- Analyzing two algebraic system is not the same type of
- F is the same see whether the domain of the V1 domain, and if f is a subset of the range of the V2 domain
- Determining whether all elements meet arithmetic operation as equal as
- F is an isomorphism proved の step V1 to V2:
- Analyzing two algebraic system is not the same type of
- See f is not bijective, f is the same whether the V1 domain of the domain, and if f is the same range as defined V2 domain
- Determining whether all elements meet arithmetic operation as equal as
- F is a homomorphism proved の step V1 to V2:
Known systems of the same type of algebraic S1 = <U, +> S2 = <V, *>, and mapping f: U-> V
| The same type of algebraic system |
Bijection can define a set of two operations on the same corresponding to each image point of the original image and calculating the number of cells |
| Algebraic System homomorphic |
f(u1+u2) = f(u1)*f(u2) |
| Homogeneous algebraic system |
and f is a bijection f (u1 + u2) = f (u1) * f (u2) |
Depending on the type of f, a different mapping may be generated:
- f is surjective, f is the algebraic system between two full homomorphism
- f is injective, f is the algebraic system between two single homomorphism
- f is injective , and the same two algebraic system, f is the algebraic system between two endomorphism
- f is a bijection f is between two algebraic systems isomorphism
- f is bijective, and the same number of two generations of systems, f is the algebraic system between two self-isomorphism
Product algebraic S1xS2 = <UxV, #>, for any <u, v> <uu, vv> belongs UxV, <u, v> # <uu, vv> = <u + uu, v * vv>
........... small n-algebraic system may generate large algebraic system
Congruence
Given <S, *>, E is the equivalence relation S, E * have substitution on the nature, i.e.,
Any x, xx, y, yy belonging to S, wherein x, xx in an equivalence class (xExx), y, yy at an equivalence class (yEyy),
Then x * y and xx * yy in an equivalence class ((x * y) E (xx * yy))
The remaining relations of gender quality:
It can be induced from a relationship with any remainder of a homomorphism
QUOTIENT: an algebraic system may generate large number of small ~ algebraic system operation equivalence classes of equivalence classes equals operator
Quotient algebra of gender quality:
An algebraic system and its algebras homomorphism, and may be configured from the operational to the natural homomorphism the number of Shang g (x) = [x] _R