Given the definition below, is that the concatenation $\emptyset A$ the same as $\lambda A,$ given $A$ a regular expression? If not, what's the difference? My guess is that if I take concatenation $AB$ as Cartesian product $A\times B,$ then suppose $A=\{0,1\}, \emptyset A=\emptyset$ but $\lambda A=A$. Is this correct?
By definition $AB = \{uv \mid u \in A, v \in B\}$. Applying this definition with $A = \emptyset$, one gets $$\emptyset B = \{uv \mid u \in \emptyset, v \in B\}= \emptyset.$$ Taking now $A = \{1\}$ (where $1$ denotes the empty word), one gets
$$\{1\}B = \{uv \mid u \in \{1\}, v \in B\} = \{1v \mid v \in B\} = B.$$