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monoid [2015/04/12 16:05]
nikolaj
monoid [2015/04/12 17:48] (current)
nikolaj
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 | @#FFBB00: definiendum | @#FFBB00: $ \langle\!\langle M,​*\rangle\!\rangle \in$ it | | @#FFBB00: definiendum | @#FFBB00: $ \langle\!\langle M,​*\rangle\!\rangle \in$ it |
 | @#AAFFAA: inclusion ​  | @#AAFFAA: $*$ ... binary operation | | @#AAFFAA: inclusion ​  | @#AAFFAA: $*$ ... binary operation |
-| @#55EE55: postulate ​  | @#55EE55: $(a*b)*c=a*(b*c)$ | +| @#FFFDDD: exists ​     | @#FFFDDD: $e$ |
-| @#FFFDDD: exists ​     | @#FFFDDD: $e\in M$ |+
 | @#55EE55: postulate ​  | @#55EE55: $e$ ... unit element $\langle\!\langle M,​*\rangle\!\rangle$ | | @#55EE55: postulate ​  | @#55EE55: $e$ ... unit element $\langle\!\langle M,​*\rangle\!\rangle$ |
 +| @#55EE55: postulate ​  | @#55EE55: $(a*b)*c=a*(b*c)$ |
  
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 The binary operation is often called //​multiplication//​ and $e$ is called the //​identity//,​ //identity element// or //unit//. The binary operation is often called //​multiplication//​ and $e$ is called the //​identity//,​ //identity element// or //unit//.
  
-One generally calls $M$ the monoid, i.e. the set where the operation "​$*$"​ is defined on. +One generally calls $M$ the monoid, i.e. the set where the operation "​$*$"​ is defined on, not the pair. For example, not that "A monoid is non-empty"​.
  
 Like above, one often uses infix notion for $*$. Like above, one often uses infix notion for $*$.
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