Differences
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monoid [2015/04/12 16:05] nikolaj |
monoid [2015/04/12 17:48] 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 $*$. | ||
