  We introduce a concept of a binary-direct implicational basis and show that
the shortest binary-direct basis exists and it is known as the $D$-basis
introduced in Adaricheva, Nation, Rand [Disc.Appl.Math. 2013]. Using this
concept we approach the algorithmic solution to the Singleton Horn Extension
problem, as well as the one set removal problem, when the closure system is
given by the canonical direct or binary-direct basis. In this problem, a new
closed set is added to or removed from the closure system forcing the re-write
of a given basis. Our goal is to obtain the same type of implicational basis
for the new closure system as was given for original closure system and to make
the basis update an optimal process.
