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Unformatted text preview: Notations: we write Z + = { , 1 , 2 , . . . } , N = { 1 , 2 , 3 , . . . } , 2 N := { 2 n ; n ∈ N } = { 2 , 4 , 6 , . . . } , and 2 N 1 := { 2 n 1; n ∈ N } = { 1 , 3 , 5 , . . . } . Properties In the following, ◦ : S × S → S is a binary operation on a nonempty set S . We write a ◦ b instead of the too formal ◦ ( a, b ). We say that ◦ is associative if ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) for all a, b, c ∈ S . This is the most important property, and all our operations below will share that. Definition. Let ◦ be an associative binary operation on a nonempty set S . Then the pair ( S, ◦ ) is a called a semigroup. (When the operation ◦ is clear from the context, we often just say that S is a semigroup.) Remark. Implicit in the word “binary operation” is the following property  often called closure : for all a, b ∈ S , a ◦ b ∈ S . We say that ◦ is commutative if a ◦ b = b ◦ a for all a, b ∈ S . [Warning: our operations are not assumed to be commutative unless explicitly stated so!] We say that ( S, ◦ ) has an identity (or “ S has an identity” for short) if there is an element e ∈ S such that e ◦ x = x and x ◦ e = x for all x ∈ S . An identity is also called a neutral element. It is easy to see that when exists, the identity is unique. [Indeed, if e and e are identities, then e = e ◦ e = e .] A semigroup with identity is sometimes called a monoid. When a semigroup ( S, ◦ ) has an identity e , we say that an element a ∈ S has an inverse (or “ a is invertible”, or “ a is a unit”) if there is a b ∈ S such that a ◦ b = e and b ◦ a = e ; it is easy to see that if exists, such an element b is unique; we usually write a 1 for this b and call it the inverse of a . [Indeed, if b and b are two such elements, then b = b ◦ ( a ◦ b ) = ( b ◦ a ) ◦ b = b .] The set of all invertible elements of S is denoted by S * . Given two binary operations + and · on the same set S , we say that · distributes over + if a · ( b + c ) = a · b + a · c and ( b + c ) · a = b · a + c · a for all a, b, c ∈ S . Examples. ( R , +), ( Z , +), ( Z + , +), ( N , · ), (2 N 1 , · ), ( Z n , +), ( Z n , · ), as well as the set of n × n real matrices with respect to matrixmultiplication are semigroups with identity. ( N , +) and (2 N , · ) are semigroups without identity....
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 Spring '08
 ctw
 Math, Binary operation, A. Theorem, m. Theorem

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