  The concept of a k-translatable groupoid is explored in depth. Some
properties of idempotent k-translatable groupoids, left cancellative
k-translatable groupoids and left unitary k-translatable groupoids are proved.
Necessary and sufficient conditions are found for a left cancellative
k-translatable groupoid to be a semigroup. Any such semigroup is proved to be
left unitary and a union of disjoint copies of cyclic groups of the same order.
Methods of constructing k-translatable semigroups that are not left
cancellative are given.
