- #1

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- TL;DR Summary
- Question is about the difference between subtraction and inverse of addition, and between division and inverse of multiplication, with respect to closure on the natural numbers

For every instance of addition or multiplication there is an inverse, closed on the naturals.

Not every instance of subtraction and division is defined, so not closed on the naturals.

This looks like two kinds of inverse.

Instance inverse - the inverse of instances of addition and multiplication operations where choosing operands for addition and multiplication always result in a natural number, as does the inverse, so the "instance inverse" is closed.

General inverse - the inverse of general addition and multiplication operations (subtraction and division) where choosing operands for subtraction and division may result in "undefined, undetermined, or disallowed" on natural numbers, so the "general inverse" is not closed.

Not every instance of subtraction and division is defined, so not closed on the naturals.

This looks like two kinds of inverse.

Instance inverse - the inverse of instances of addition and multiplication operations where choosing operands for addition and multiplication always result in a natural number, as does the inverse, so the "instance inverse" is closed.

General inverse - the inverse of general addition and multiplication operations (subtraction and division) where choosing operands for subtraction and division may result in "undefined, undetermined, or disallowed" on natural numbers, so the "general inverse" is not closed.