I need some help of the logic experts. I would like to evaluate the following signatures $\sigma$, such that $|\sigma^{Op}|=2$ and $t$ is a $\sigma$ term. Sometimes there are no solutions and sometimes more.
The countable set $\mathbb X=\{x^0,x^1,x^2,...\}$ (the numbers should be above, I do not know how to do that in latex) of symbols is called variable.
A signature is a pair $\sigma=(\sigma^{Op},ar) $ where $ar:\sigma^{Op}\rightarrow \mathbb N$ which gives the arity of the symbols and the elemnts of $\sigma^{Op}$ are operation symbols.
I have no idea how to evaluate the signatures, may you could show me that. For me it is also not not clear what this numbers above the variables exactly mean, may you have an idea?
Maybe the following defintion might be helpful: The set of all $\sigma$ terms is the smallest language over the alphabet $\sigma^{Op}\cup X$ which satisfy
(i) Every variable is a $\sigma$ term.
(ii) If $f\in\sigma^{Op}$ and $k=ar(f)$ and $t_1,...,t_k$ are $\sigma$ terms, then $ft_1...t_k$ is a $\sigma$ term