For formal grammars $𝐴$ and $𝐵$, the residual $𝐴⊸𝐵$ (sometimes called the lollipop or linear implication) is the right adjoint to the grammar tensor $𝐴\otimes 𝐵$. It describes strings that, when prefixed by a string matching $𝐴$, will match $𝐵$.

Formally, it is defined as:

Intuitively, a parse of $𝐴⊸𝐵$ at a string $𝑣$ is a function that takes any prefix string $𝑢$ and a parse of $𝑢$ in $𝐴$, and produces a parse of the full concatenated string $𝑢·𝑣$ in $𝐵$.