Common Properties of Algebraic Systems
Ring and Fields: (R, + ,* ) |
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| + | * | ||
| Associative | (a+b)+c=a+(b+c) | (a*b)*c=a*(b*c) | Ring properties are in yellow |
| Distributive | a*(b+c)=a*b+a*c |
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| Commutative | a+b=b+a | a*b=b*a | (Commutative Ring) |
| Identity | a+e=a | a*e=a=e*a | (Ring with Unity) |
| Inverse | a+x=e | a*x=e x<>e | (Field) |
a,b,e are in the set of R.
+ and * are binary operators.
Other common operators are (^,V), (AND, OR).
The inverting element is often written as: a-1, -a, f -1,
f -1(x), 1/a.
e is sometimes called the neutral element.
Monoids and Groups: (R, +) |
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| + | Semi-Group | Monoid | Group | Commutative Group | |
| Associative | (a+b)+c=a+(b+c) | X | X | X | X |
| Identity | a+e=a | X | X | X | |
| Inverse | a+x=e | X | X | ||
| Commutative | a+b=b+a | X | |||
+ stands for any operator such as * and +.
e is the identity element. .
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