1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is a semigroup

  1. Jul 23, 2014 #1
    Definition/Summary

    A semigroup is a set S with a binary operation S*S -> S that is associative.

    A semigroup with an identity element is a monoid, and also with an inverse for every element is a group.

    A semigroup may have idempotent elements, left and right identities, and left and right zeros (absorbing elements).

    Equations

    Associativity: [itex]\forall a,b,c \in S ,\ (a \cdot b) \cdot c = a \cdot (b \cdot c) [/itex]

    Idempotence: [itex]a \cdot a = a[/itex]
    Left identity e: [itex]\forall a \in S,\ e \cdot a = a [/itex]
    Right identity e: [itex]\forall a \in S,\ a \cdot e = a [/itex]
    Left zero z: [itex]\forall a \in S,\ z \cdot a = z [/itex]
    Right zero z: [itex]\forall a \in S,\ a \cdot z = z [/itex]

    Extended explanation

    If a semigroup has both left and right identities, then they are a unique two-sided identity.

    If e1 is a left identity and e2 is a right identity, then e1*e2 = e1 by e2 being a left identity, but e1*e2 = e1 by e2 being a right identity. These two equations imply that e1 = e2 = e. If there is more than one possible left or right identity, then this argument shows that they are all equal to e.

    If a semigroup has both left and right zeros, then they are a unique two-sided zero. The proof closely parallels that for identities. For left zero z1 and right zero z2, z1*z2 = z1 by the left-zero definition and z1*z2 = z2 by the right-zero definition.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: What is a semigroup
  1. WHat is this? (Replies: 9)

  2. What is ^? (Replies: 2)

  3. What is a Lagrangian? (Replies: 7)

  4. What's the connection? (Replies: 3)

Loading...