# Quasigroups extracted from the rational numbers and division

Consider a quasigroup $(Q,/)$, that is, Q is a set and for $\forall a,b\in Q$ there are unique solutions to the equations $x/a=b$ and $a/y=b$. How to find a maximal set of independent representants of identities of expressions formed by matching brackets, the sign / and variables, like

1. a/(b/c)=c/(b/a)
2. (a/b)/c=(c/b)/a
3. a/a=b/b
4. a/(b/b)=a

for all variables $a,b,c,\dots \in Q$ such that the identities are valid in the quasigroup $(Q_+,/)$ of positive rational numbers? Or more general, so that the identities are valid in the quasigroup associated with a certain group?

A representant of an identity is a string of signs with matching brackets, and a single '=' placed in the string so that the expressions on the right and on the left of '=' represent elements in the quasigroup.

A set of representants of identities is independent if it isn't possible to derive one of the identities from the other by modus ponens or substitution:

”The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation ’immediate consequence’, i.e. formula c of a and b is defined as an immediate consequence in terms of modus ponens or substitution.” (Gödel)

What branches of mathematics are suitable for studying problems like this?

• For any group G, you could make the same definition. I guess there is nothing special about $\mathbb Q_+$ which makes it different from any other torsion-free commutative group (from the point of view of the identities which hold in the quasigroup defined from it). Separate point: I guess the question could be improved by making more precise what "independent" means. – Hugh Thomas supports Monica Oct 15 '15 at 2:10