symmetric monoidal (∞,1)-category of spectra
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
For $A$ an abelian group and $(a,b) \in A \times A$ a pair of elements, their difference is the element
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
Specifically if $V = \mathbb{R}$ is the real line or the rational numbers of just the integers, one just writes $y-x$. Etc.
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
$a$ 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.
A derivative is a limiting ratio of differences.