SEMIGROUP PROPERTIES OF COOPERATIONS ON FINITE SETS

Let A^⊔n := {1, …, n} × A be the n-th copower of the set A. An n-ary cooperation is a mapping f : A → A^⊔n. If f is an n-ary cooperation and if g₁, …, g_n are k-ary cooperations on A, we may define a composition and obtain a new k-ary cooperation defined on A. Together with the injection cooperations defined by a ↦ (i, a) for 1 ≤ i ≤ n, all cooperations defined on A form a multibased algebra. This algebra is called the clone of all cooperations defined on A. If we define by a binary operation on the set of all n-ary cooperations, the set of all n-ary cooperations defined on A forms a semigroup with respect to this operation. We determine the order of the elements of this semigroup, characterize all idempotent and regular elements, ask for bands of n-ary cooperations and characterize Green's relations and .