Consider the set of integers, Z. A binary operation ⊗ on Z is called a "pseudo-multiplication" if it satisfies all of the three properties:
⊗ is commutative: m ⊗ n = n ⊗ m,
⊗ is associative: (m ⊗ n) ⊗ p = m ⊗ (n ⊗ p), and
⊗ is distributive over + (the usual addition of integers): m ⊗ (n + p) = m ⊗ n + m ⊗ p.
Find all possible pseudo-multiplication operations ⊗ on Z. For each operation you find, prove it has the three properties so you know it really is a pseudo-multiplication.
If you don't find all, find at least one (besides ·) and find as many as you can.
(This is part of the project.) For each pseudo-multiplication you found, answer the following questions if you can:
Does it have a pseudo-multiplicative identity element?
Does it have the cancellation property?
Find all integers that have pseudo-multiplicative inverses.