transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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.
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(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
In a co-Heyting algebra, subtraction is the operation left adjoint to the join operator:
This is related to subtractive logic.
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