SEMIGROUP PROPERTIES OF COOPERATIONS ONFINITE SETS

摘要

Let A" := {1, , n} x A be the n-th copower of the set A. An n-ary cooper-ation is a mapping f : A AUn. If f is an n-ary cooperation and if gi, . , gnare k-ary cooperations on A, we may define a composition comp (f, gi, , gn)and obtain a new k-ary cooperation defined on A. Together with the injection cooperations tin'A : A Alin defined by a (i, a) for 1 < i < n, all coopera-tions defined on A form a multibased algebra. This algebra is called the cloneof all cooperations defined on A. If we define by f g := comp:1(f , g, . . . , g) abinary operation on the set of all n-ary cooperations, the set of all n-ary co-operations defined on A forms a semigroup with respect to this operation. Wedetermine the order of the elements of this semigroup, characterize all idempo-tent and regular elements, ask for bands of n-ary cooperations and characterizeGreen's relations G and R.