nLab difference

Definition

For abelian groups

For $A$ an abelian group and $(a,b) \in A \times A$ a pair of elements, their difference is the element

$$b - a \coloneqq b + a^{-1} \in A \,.$$

This is often considerd for the case that $A$ is the abelian group underlying a vector space $V$, which which case one typically denotes the elements instead as $(\vec x, \vec y) \in V \times V$ and their difference as

$$\vec y - \vec x \in V \,.$$

Specifically if $V = \mathbb{R}$ is the real line or the rational numbers of just the integers, one just writes $y-x$. Etc.

For commutative semigroups and magmas

For $(A,+)$ a commutative semigroup or magma and $(a,b) \in A \times A$ a pair of elements, their difference, if it exists, is an element $c \in A$ such that

$$c + a = b \,.$$

$a$ is called the subtrahend of $b$ and $b$ is called the minuend of $a$.

For example, in the ordered commutative semigroup of positive integers $\mathbb{N}^+$, two positive integers $m,n \in \mathbb{N}^+$ have a difference $n - m$ if $m \lt n$.

If the magma is a multiplicative monoid $\cdot$ of a commutative ring, then the difference is usually called a quotient (see divisor (ring theory).

A commutative quasigroup is a commutative magma such that every pair of elements $(a,b) \in A \times A$ has a difference.

Related concepts

• subtraction

• monus

• A derivative is a limiting ratio of differences.