Abstract
One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups
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