Notation seems to vary a bit on here, so for this question:
$N$ is the set of nonterminals $\{S, A\}$,
$T$ is the set of terminals $\{a, b, c\}$
$P$ is the set of productions $u \rightarrow v$.
I have to determine the type of grammar of $G$, for $G = (\{S, A\}, \{a, b, c\}, S, \{S \rightarrow Sc, S \rightarrow Ac, A \rightarrow ab, A \rightarrow aAb\}$).
I think it is a linear grammar. I know that for a linear grammar, each $u \rightarrow v \in P$ has $u \in N$ and $v \in T^{*} \cup T^{*}NT^{*}.$
The left side of each production is a nonterminal, so that's okay. I just have to check the right side:
$Sc = \lambda Sc \in T^{*}NT^{*}$,
$Ac = \lambda Ac \in T^{*}NT^{*}$,
$ab \in T^{*}$,
$aAb = T^{*}NT^{*}$.
Since each $v$ is in either the set $T^{*}$ or $T^{*}NT^{*}$, I think this is a linear grammar that generates a linear language.
Is there anything about this that is incorrect? I just started studying grammars and I want to make sure I am understanding it. Please advise!
The language is $L = \{a^nb^nc^m\mid m,n\geq 1\}$. This language is contextfree not regular. It can be proved by the pumping lemma.