Examples
| set | operation | identity |
|---|---|---|
| real numbers | + (addition) | 0 |
| real numbers | · (multiplication) | 1 |
| non-negative numbers | ab (exponentiation) | 1 (right identity only) * |
| integers (to extended rationals) | ||
| positive integers | least common multiple | 1 |
| non-negative integers | greatest common divisor | 0 (under most definitions of GCD) |
| m-by-n matrices | + (addition) | matrix of all zeroes |
| n-by-n square matrices | matrix multiplication | In (matrix with 1 on diagonal and 0 elsewhere) |
| m-by-n matrices | (Hadamard product) | Jm, n (Matrix of ones) |
| all functions from a set M to itself | ∘ (function composition) | identity function |
| all distributions on an group G | ∗ (convolution) | δ (Dirac delta) |
| strings, lists | concatenation | empty string, empty list |
| extended real numbers | minimum/infimum | +∞ |
| extended real numbers | maximum/supremum | −∞ |
| subsets of a set M | ∩ (intersection) | M |
| sets | ∪ (union) | ∅ (empty set) |
| a boolean algebra | ∧ (logical and) | ⊤ (truth) |
| a boolean algebra | ∨ (logical or) | ⊥ (falsity) |
| a boolean algebra | ⊕ (Exclusive or) | ⊥ (falsity) |
| knots | knot sum | unknot |
| compact surfaces | # (connected sum) | S2 |
| only two elements {e, f} | ∗ defined by e ∗ e = f ∗ e = e and f ∗ f = e ∗ f = f |
both e and f are left identities, but there is no right identity and no two-sided identity |
* assuming 00 = 1, or zero has to be excluded from the domain.
Read more about this topic: Identity Element
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