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Semigroup

# 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
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?
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.
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.
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
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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