# What is a semigroup

1. Jul 23, 2014

### Greg Bernhardt

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: $\forall a,b,c \in S ,\ (a \cdot b) \cdot c = a \cdot (b \cdot c)$

Idempotence: $a \cdot a = a$
Left identity e: $\forall a \in S,\ e \cdot a = a$
Right identity e: $\forall a \in S,\ a \cdot e = a$
Left zero z: $\forall a \in S,\ z \cdot a = z$
Right zero z: $\forall a \in S,\ a \cdot z = z$

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!