Let there be a set P, and a set K such that $P\subset K$. Let there be 2 binary operations closed on K written $+$ and $\times$. Is there any way to define K as having only elements composed of applications of these operations on elements of P?
In other words, let $P = \{A,B,C\}$. K would then be defined as the set $K =\{A+A,A\times B,A+B\times C,\cdots\}$ and all other possible combinations of the operations on P.
The set $K$ is typically called the closure of $P$ under $+,\times$. We can define it recursively by setting $K_0=P$ and, given $K_n$, letting $K_{n+1}$ be $K_n\cup\{a+b\mid a,b\in K_n\}\cup\{a\times b\mid a,b\in K_n\}$. Finally, $K=\bigcup_n K_n$.
The same idea applies to any number of operations, regardless of whether they are binary or not. Usually, given a set $S$ closed under the operations, we would define the set $K$ you are interested in as the closure of $P$ inside $S$. And even more generality is possible, leading for instance to the idea of Skolem hulls in first order logic.