# 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 $s:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$, where, of course, $s(m,n)=m-n$.

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

## Definition

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

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

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

This is related to subtractive logic.

## Examples

• Subtraction in the commutative monoid of the natural numbers $\mathbb{N}$, is only partially defined. It is given by the monus/truncated subtraction operator:
$\dot - \; : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}.$
$- \; :\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}.$