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Sufficient conditions for the regularity of a context-free language

Let $M$ be a Monoid and, for any subsets $A, B ⊆ M$ define the left-quotient $$A \backslash B = \{ m ∈ M: (∃a ∈ A) am ∈ B\}.$$ For $m ∈ M$ and $B ⊆ M$, define $m \backslash B = \{m\} \backslash B$. ...
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