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semigroup

 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: $\forall a \in S,\ a \cdot z = z$

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 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.

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