Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows:
- $a \boxplus b=a+b-ab$ for all $a,b \in F$.
- $a \boxdot b=1-t^{\log_t (1-a) \log_t (1-b)}$ for all $a,b\in F$
For which values of $t$ does $F$, together with these operations, form a field?
I'm stuck on finding the possible $t$ values for this. Any help with the $t$ values is greatly appreciated.
As an example, let's check distributivity.
$\begin{align*} a\boxdot(c\boxplus d)&=a\boxdot(c+d-cd)\\ &=1-t^{\log_t(1-a)\log_t(1-(c+d-cd))}\\ \end{align*}$
$\begin{align*} (a\boxdot c)\boxplus (a\boxdot d)&=(1-t^{\log_t(1-a)\log_t(1-c)})\boxplus (1-t^{\log_t(1-a)\log_t(1-d))})\\ &= 1-t^{\log_t(1-a)\log_t(1-c)}+1-t^{\log_t(1-a)\log_t(1-d))}-(1-t^{\log_t(1-a)\log_t(1-c)})(1-t^{\log_t(1-a)\log_t(1-d))})\\ &= 1-t^{\log_t(1-a)\log_t(1-c)}-t^{\log_t(1-a)\log_t(1-d))}\\ &+t^{\log_t(1-a)\log_t(1-c))}+t^{\log_t(1-a)\log_t(1-d))}-t^{\log_t(1-a)\log_t(1-c)+ \log_t(1-a)\log_t(1-d)}\\ &=1-t^{\log_t(1-a)\log_t(1-c)+ \log_t(1-a)\log_t(1-d)}\\ &=1-t^{\log_t(1-a)\log_t[(1-c)(1-d)]} \end{align*}$
These expressions match, so they don't give any restriction on $t$ (except that $t>0$ in order for $\log_t$ to be defined.) Now check the other axioms similarly.