nLab 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.

Abstract

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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, which are models of subtractive logic, subtraction is the operation left adjoint to the join operator:

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

Properties

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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:

$$\dot - \; : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}.$$

• Subtraction in the abelian group of the integers $\mathbb{Z}$, is well known to be an entire relation on the integers,

$$- \; :\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}.$$

• Subtraction in a boolean ring is the symmetric difference, and is the same as addition due to a boolean ring having a characteristic of 2.

References

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Related concepts

• difference