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Semigroup - Algebraic Systems and Groups Lecture 13...

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Algebraic Systems and Groups Lecture 13 Discrete Mathematical Structures
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Algebraic Operations Function ƒ: A n B is called an n -nary operation from A to B . Binary operation: ƒ: A × A B (ƒ: A × A A) An example: a new operation “*” defined on the set of real number, using common arithmetic operations: x*y = x+y-xy Note: 2*3 = -1 A 0.5*0.7 = 0.85
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Closeness of Operations For any operation ƒ:A n B, if B A, then it is said that A is closed with respect to ƒ. Or, we say that ƒ is closed on A . Example: Set A={1,2,3,…,10}, gcd is closed, but lcm is not.
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Operation Table Operation table can be used to define unary or binary operations on a finite set (usually only with several elements) * a b c d a b c c*b d How many binary operations can be defined here?
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Association Operation “°” defined on the set A is associative if and only if: For any x, y, z A, (xAy)Az = xA(yAz) If “ ± ” is associative, then x 1 * x 2 * x 3 * …* x n can be computed by any order of among the ( n -1) operations, with the constraint that the order of all operands are not changed.
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Commutation Operation “°” defined on the set A is associative if and only if: For any x, y A, xAy = yA x If “² ” is commutative and associative, then x 1 * x 2 * x 3 * …* x n can be computed by any order of the operations, and in any permutation of all operands.
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Distribution Two different operations must be defined for an algebraic system for discussion of distribution. Operation “]” is distributive over “ ” (both operations defined on the set A ) if and only if : For any x, y, z A, xA(y z) = (xAy) (xAz) (Exactly speaking, this is the first distributive property)
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Identity of an Algebraic System For arithmetic multiplication on the set of real number, there is a specific real number 1 A satisfying that for any real number x , 1∙x=x∙1=x An element e is called the identity element of an algebraic system (S,² ) if and only if : For any x S, e¤x=x “ e=x A Denotation: 1 S , or simply 1, but remember that it is not that “1”. It is not that every algebraic system has its identity element.
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Left Identity and Right Identity e l is called a left identity of an algebraic system S , if and only if: For any x S , e l o x = x Right identity e r * can be defined similarly. * a b c d a a d c a b b d c b c c d c c d d d b d * a b c d a a b c d b a b c d c a b c d d a b c d
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More about Identity For any algebraic system S : There may or may not be left or right identity. There may be more than one left or right identities. If S has a left identity and a right identity as well, then they must be equal, and this element is also an identity of the system: e l = e l Be r = e r If existing, the identity of an algebraic system is unique: e 1 = e 1 A e 2 = e 2
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Inverse (Inverse can be discussed for those system with identity.) For a given element x in the system S , if there is some element x ’ in the system, satisfying that x ’A x =1 S , then x ’ is called a left inverse of x.
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