nLab
difference

Contents

Context

Algebra

Arithmetic

Contents

Definition

For abelian groups

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

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

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

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

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

For commutative semigroups and magmas

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

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

aa is called the subtrahend of bb and bb is called the minuend of aa.

For example, in the ordered commutative semigroup of positive integers +\mathbb{N}^+, two positive integers m,n +m,n \in \mathbb{N}^+ have a difference nmn - m if m<nm \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)A×A(a,b) \in A \times A has a difference.

Last revised on May 19, 2021 at 13:25:35. See the history of this page for a list of all contributions to it.