# Identity Element - Examples

Examples

set operation identity
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
ee = fe = e and
ff = ef = 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.