**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 | I_{n} (matrix with 1 on diagonaland 0 elsewhere) |

m-by-n matrices |
(Hadamard product) | J_{m, 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 bye ∗ e = f ∗ e = e andf ∗ 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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