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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

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