  In this work we characterize shift spaces over infinite countable alphabets
that can be endowed with an inverse semigroup operation. We give sufficient
conditions under which zero-dimensional inverse semigroups can be recoded as
shift spaces whose correspondent inverse semigroup operation is a 1-block
operation, that is, it arises from a group operation on the alphabet. Motivated
by this, we go on to study block operations on shift spaces and, in the end, we
prove our main theorem, which states that Markovian shift spaces, which can be
endowed with a 1-block inverse semigroup operation, are conjugate to the
product of a full shift with a fractal shift.
