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Consider the structure $(\mathbb{Z}, *)$. Are there any congruence relations on that structure, in the sense of universal algebra, that are not of the form mod n for some integer n? In fact, is there a classification of all the congruence relations on that structure?

user107952
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    How about $a\sim b$ when $a$ and $b$ "have the same sign"? – Angina Seng Jun 13 '20 at 22:28
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    @RobArthan I think you misunderstand. I think Angina is referring to the relation, "a and b are both zero or both positive or both negative" – user107952 Jun 13 '20 at 22:38
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    Another example is the relation "$a \sim b$ iff $\frac{a}{b}$ is a nonzero rational square or $a=b=0$". Yet another example is the relation of having the same absolute value. – Geoffrey Trang Jun 13 '20 at 22:39
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    Mea culpa for a dumb comment and a dumb answer. The OP needs to tidy up the question: "mod n for some n" isn't very clear. – Rob Arthan Jun 13 '20 at 22:49
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    What is the operation? $*$ would normally be understood to be multiplication, in which case, are you looking at monoid congruences? – Arturo Magidin Jun 13 '20 at 23:49
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    It's a bit easier to think about $\mathcal{N}=(\mathbb{N}_{>0};\times)$ first - that already has lots of additional congruences, and it's much snappier to describe structurally (it's the free monoid on $\aleph_0$-many generators). Every congruence of $\mathcal{N}$ lifts to a congruence of $\mathcal{Z}=(\mathbb{Z};\times)$ via the $\mathcal{Z}$-congruence ${(x,y):\vert x\vert=\vert y\vert}$. – Noah Schweber Jun 16 '20 at 23:58

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