Examples
- The most well known example of an absorbing element in algebra is multiplication, where any number multiplied by zero equals zero. Zero is thus an absorbing element.
- Floating point arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is absorbing element for every operation, ie. x+NaN=NaN+x=NaN, x-NaN=NaN-x=NaN etc.
- The set of binary relations over a set X, together with the composition of relations forms a monoid with zero, where the zero element is the empty relation (empty set).
- The closed interval H= with x∘y=min(x,y) is also a monoid with zero, and the zero element is 0.
- More examples:
| set | operation | absorber |
|---|---|---|
| real numbers | · (multiplication) | 0 |
| nonnegative integers | greatest common divisor | 1 |
| n-by-n square matrices | · (multiplication) | matrix of all zeroes |
| extended real numbers | minimum/infimum | −∞ |
| extended real numbers | maximum/supremum | +∞ |
| sets | ∩ (intersection) | { } (empty set) |
| subsets of a set M | ∪ (union) | M |
| boolean logic | ∧ (logical and) | ⊥ (falsity) |
| boolean logic | ∨ (logical or) | ⊤ (truth) |
Read more about this topic: Absorbing Element
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