Understanding the meaning of $\phi :G\times X\to X$ This is a very simple thing:
$$\phi : G\times X\to X$$
but I am having trouble understanding a few aspects. Specifically:


*

*Does it mean $\phi : (G \times X) \to X$ or $\phi : G \times (X \to X)$, or are both equivalent or similar in some sense. The order of operations is confusing me.

*What does a specific invocation of the function look like? Is it $\phi (g,x) \mapsto y \in X$, or $\phi (g) \mapsto (x,y)$, or how does it look?

 A: Syntactically, only one interpretation is possible.

Name of function $\color{red}\colon$ domain of function$\color{red}\to$ range of function

where domain and range are sets.
Thus given an element $(g,x)$ of $G\times X$ (that is, given an element $g$ of $G$ and an element $x$ of $X$), $\phi$ produces an element of $X$, which we denote as $\phi(g,x)$.
A: The correct operation you are looking for is called $currying$ and it specifies $f:A\times B\rightarrow C$ has a representation of the form $\bar{f}:A\rightarrow BfC$, where $BfC$is the collection of functions from subsets of $B$ into subsets of $C$.
For example, when you have an operation like addition of integers, wht you are defining is functors $+x$ such that when you apply them to the partial order of integers, you get the following result $+x(n)=n+x$. This means that the operation $+$ in $\mathbb Z$ is defined by $+x(n)=n+x$. For example, the functor $+4$ can be applied to the arrow $2<5$ and this gives the arrow $6<9$.
What we are saying is that instead of looking at operations as functions that map ordered sets into a third set, we can visualize operations as functions that map sets into a function space. In other words, an operation is a function whose image is a space of functions.
