Given two prefix dosed languages K, L subset-or-equal-to SIGMA*, where K subset-or-equal-to L represents the desired closed-loop behavior and L is the open-loop behavior, there exists a finite-state supervisor that enforces K in the closed loop if and only if there is a regular, prefix-closed M subset-or-equal-to SIGMA*, such that : 1) MSIGMA(u) and L subset-or-equal-to M, and 2) M and L = K (cf. [9] for an equivalent definition). In this paper, we show that this is equivalent to : 1) the controllability of sup{P subset-or-equal-to K or LBAR pr(P) = P} with respect to SIGMA*, and 2) the regularity of sup(P subset-or-equal-to K or LBAR pr(P) = P}, where LBAR = SIGMA* - L and pr(.) is the set of prefixes of strings in the language argument. We use this property to investigate the issue of deciding the existence of a finite-state supervisor for different representations. We also present some properties of the language sup{P subset-or-equal-to K or LBAR pr(P) = P}, along with implications to the synthesis of solutions to the supervisory control problem with the fewest states.