If I look at the Peano axioms for arithmetic, it's clear to me how they define the addition and multiplication of every natural number with every other natural number.
Then if I look at the two element Boolean algebra, I can see again how the axioms define the truth tables for $\land$, $\lor$ and $\lnot$, giving complete information for every unary and binary operation on the set $\{T, F\}$.
Now consider a set $\Omega$ with cardinality $n$, and look at the Boolean algebra of $\cap, \cup$ and ${}^c$ (complement in $\Omega$) on the power set $2^\Omega$. It's clear to me how the definition $X\cap Y = \{x \in \Omega | x \in X \land x \in Y\}$ can be used to give the full multiplication table for $\cap$, and similarly for $\cup$.
Is it possible to give the full multiplication table for $\cap$ and $\cup$ in $2^\Omega$ just from the axioms of the Boolean algebra of these operations? I can see how the first and last rows and columns of the multiplication tables (for $\Omega$ and for the empty set) are defined by the axioms, and how the diagonals are defined by the axioms. It seems to me that the rest of the multiplication tables are not defined by the axioms. Is this true? By analogy from the Peano axioms and the two element Boolean algebra I expected them to be, so I'm surprised that they appear not to be.
Should I expect the axioms of an algebra to define the full multiplication tables of its operations in general?