Given any monoid (that is a set $A$ equipped with an associative operation $\cdot$ and an identity $1$), we can define "finite products" roughly by:
$$\prod_{i=1}^n a_i = a_1\cdot a_2\cdot \dots \cdot a_n$$
where $\prod_{i=1}^0 = 1$.
Possible monoids are for example $(\mathbb{R},\cdot, 1)$ yielding "$\prod$", $(\mathbb{R},+,0)$ yielding "$\sum$" or $(P(S), \cup, \emptyset)$ yielding "$\bigcup$" and so on and so forth.
So, we can also view a monoid as a set $A$ together with a map $A^* \to A, (a_n) \mapsto \prod_{i=1}^n a_i$ taking lists (words, tuples) of elements of $A$ to elements of $A$.
Occassionally however, we may find "maps" (broadly speaking) which not only accept finite lists, but also infinite lists or even bigger families of elements as objects.
For example, a complete lattice is a set $A$ equipped with maps $\bigvee$ and $\bigwedge$ taking abitrary families of elements of $A$ to elements of $A$.
Intuitively, if you take the set of all "small sets" (this is usually realized as a proper class) as the set $A$, then you get a complete lattice with operations $\bigcup$ and $\bigcap$ called union and intersection, which take families of elements of $A$ (that is sets of sets) to sets.
Best Answer
Books usually define the notations before they use them, and this one is no different.