Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
monoid [2015/03/28 20:17]
nikolaj
monoid [2015/04/12 17:48]
nikolaj
Line 4: Line 4:
 | @#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 |
 +| @#FFFDDD: exists ​     | @#FFFDDD: $e$ |
 +| @#55EE55: postulate ​  | @#55EE55: $e$ ... unit element $\langle\!\langle M,​*\rangle\!\rangle$ |
 | @#55EE55: postulate ​  | @#55EE55: $(a*b)*c=a*(b*c)$ | | @#55EE55: postulate ​  | @#55EE55: $(a*b)*c=a*(b*c)$ |
-| @#FFFDDD: exists ​     | @#FFFDDD: $e\in M$ | 
-| @#FFFDDD: for all     | @#FFFDDD: $a\in M$ | 
-| @#55EE55: postulate ​  | @#55EE55: $a*e=e*a=a$ | 
  
 ----- -----
Line 13: Line 12:
 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 $*$.
Line 21: Line 20:
  
 ----- -----
 +=== Requirements ===
 +[[Unit element]]
 === Subset of === === Subset of ===
 [[Semigroup]] [[Semigroup]]
Link to graph
Log In
Improvements of the human condition