nLab
subtraction

Subtraction

Idea

The basic example of subtraction is, of course, the partial operation in the monoid of natural numbers or in the integers. It is often the first illustration of a non-associative operation met in abstract algebra. We think of subtraction as an operation s:×s:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}, where, of course, s(m,n)=mns(m,n)=m-n.

There are numerous related abstractions of this, relating to different aspects of the basic operation.

Abstract

Definition

  1. (Universal Algebra) In the sense of Ursini in the context of a varietal theory, a subtraction term, ss, is a binary term ss satisfying s(x,x)=0s(x, x) = 0 and s(x,0)=xs(x, 0) = x. (see subtractive variety

  2. In a co-Heyting algebra, subtraction is the operation left adjoint to the join operator:

    (y)(y) (- \setminus y) \dashv (y \vee -)

This is related to subtractive logic.

Properties

Examples

˙:×.\dot - \; : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}.
:×.- \; :\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}.

References

Last revised on May 24, 2017 at 10:16:15. See the history of this page for a list of all contributions to it.