structures

structures - Notations we write Z = 1 2 N = 1 2 3 2 N:= 2 n...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 matrix-multiplication are semigroups with identity. ( N , +) and (2 N , · ) are semigroups without identity....
View Full Document

{[ snackBarMessage ]}

Page1 / 4

structures - Notations we write Z = 1 2 N = 1 2 3 2 N:= 2 n...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online