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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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 Breakdown Mathematics > Algebra >> Group Theory