I’ve been learning on *Formal Languages and Automata* of Peter Linz(6th edition).

in the chapter 3 of this book, it explains the **primitive regular expression**.

but I don’t know what is the difference between $ \phi$ and $ \lambda$ .

of course, I know $ \lambda$ means the empty string, so that $ \lambda s=s\lambda$ .

and the textbook explains the meaning of $ \phi$ is the empty set.

and more, the textbook explains that $ \phi$ can be accepted by a deterministic finite automata $ \left< Q, \Sigma, \delta, q_0 , F \right>$ in which $ Q=\{ q_0, q_1 \}$ , $ \forall a \in \Sigma:\delta(q_0,a)\text{ is not defined}$ , and $ F=\{q_1\}$ .

so, I guess the meaning of the $ \phi$ is the **rejected string**.

but How can the expression $ (\phi *)*$ mean $ \lambda$ ?

and what’s the meaning of the expression $ a\phi$ ?