I am studding many-sorted algebra. In this paper (page 4), it is clearly said that in the signature $(S, \leq , \Sigma)$, $\Sigma$ is a family $\Sigma=\{ \Sigma_{w,s}\}_{(w,s)\in S^*\times S}$ of (not necessarily disjoint) sets of operator symbols. Also we have a similar thing in this one (page 6).
My Questions:
1) Does this means that for one operator symbol, say $f$, we can assign more than one Arity schema? (i.e., having one function symbols with different arities). In more detail, for example, I can have $f: A_1 \rightarrow B_1$, and also $f:A_1 \times A_2 \rightarrow B_1$. ( or I am missing something )
2) If answer to the question one is "yes", is this allowed in other areas of algebraic studies as well.
The answer to Question 1 is yes. The intention is that the same operator symbol can be used for (what I would call) several operators, of distinct arities. This convention is quite unusual. In principle, it would not cause problems if the same operation symbol $f$ is used with two (or more) arities that differ as to the inputs of $f$, because then one can infer the intended meaning of $f$ by looking at the arguments to which it is applied. But things look bad (to me) if $f$ has two arities that involve the same input sorts but different output sorts. Then, if $t_1,\dots,t_k$ have the appropriate sorts to serve as inputs of $f$, then $f(t_1,\dots,t_k)$ would have two meanings, of different sorts.