I'm wondering if the rationals between $0$ and $1$, have been studied in a systematic manner using abstract algebra.
Is there any interesting theory behind this set?
2026-04-05 02:01:12.1775354472
Is there an abstract algebraic way to analyze the rational numbers between 0 and 1 (inclusive)?
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Lattice theory is studied algebraically, and this is a natural example of a bounded totally ordered lattice, so I would say "yes."
If you are thinking explicitly of some operation induced by that of $\Bbb Q$, I can think of no better suggestion than what user73985 has already offered with $\Bbb Q/\Bbb Z$. The rationals in $[0,1)$ act as coset representatives. Although $1$ and $0$ overlap here, it's still an excellent suggestion.