subtraction (changes) in nLab

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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:ℤ×ℤ→ℤs:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}, where, of course, s(m,n)=m−ns(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, 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, which are models of subtractive logic, subtraction is the operation left adjoint to the join operator:

   (−∖y)⊣(y∨−) (- \setminus y) \dashv (y \vee -)

Properties

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Examples

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

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

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

−:ℤ×ℤ→ℤ.- \; :\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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• difference