- ("Tending to substitute or switch"): the ability to change the order of something without changing the result.
- Example: 4+8 = 8+4
- Within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed. If an operation is non-associative, it may be left- or right-associative
- Example: (1+2)+3 = 1+(2+3), but (2-1)-1 ≠ 2-(1-1)
- Example: 4 • (2 + 3) = (4 • 2) + (4 • 3)
- A set is said to be closed under some operation if the operation on members of the set produces a member of the set.
- Example: the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not.
- Existence of identity element
- Example: 1 • x = x and 0 + x = x
- Existence of inverse
- Example: x • x^-1 = 1 and x + (-x) = 0
- An additive function is a function that preserves the addition operation.
- Example: T(u+v) = T(u) + T(v)
- A homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power (often 1) of this factor.
- Example: T(a • v) = a • T(v)