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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!
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!