On the set $S = \{0, 1 \}$ let $f : S^3 \mapsto S$. Ternary operator $f(a, b, c)$ evaluates to $0$ when at least two arguments equal $0$; otherwise it returns $1$. One can form various terms in a symbolic operation $f$. An example: $f'(a, b, c, d, e) = f(a, f(b, c, d), e)$ and then evaluate these in the algebra $(S, f)$ to get operations on $S$ derived from $f$.
The thing which is not quite obvious, is how one might "derive" operations from $f$? Say, I want "to derive" an unary or binary operation (no matter what they will do, just the fact that none would be a $S^3 \mapsto S$ mapping). The only way I see to achieve that seems to be insane: I could factor $g(a, b)$ from $f(a, b, c)$ by... ignoring $c$. But it sounds so impractical, that I can't believe such a nonsense should be considered as a valid solution.
So, probably, there exist some another way "to derive" non-$f$ operations on the set $S$? Or did I misunderstood the entire "to derive" concept?