Semigroup Properties of N-Ary Operations on Finite Sets

Abstract

A clone is a set of operations defined on a base set A which is closed under composition and contains all the projection operations. There are several ways to regard a clone as an algebraic structure (see e.g. [3]). If f, g1,…,gn : An → A are n-ary operations defined on A, then by Sn(f, g1 … , gn)(a1 … , an) := f(g1(a1,…,an),…,gn(a1,…,an)) for all a1,…, an ∈ A an (n + 1)-ary operation on the set On(A) of all n-ary operations can be defined. From this operation one can derive a binary operation + defined by f + g := Sn(f, g,…,g) and obtains a semigroup (On(A);+). The collection of all clones of operations on a finite set forms a complete lattice. This lattice is well-described ([4], [5]) if |A| = 2. If |A| > 2, this lattice is uncountably infinite and very complex. In this paper instead of clones we study semigroups of n-ary operations, i.e. subsemigroups of the semigroup (On(A); +) and their properties. We look for idempotent and regular elements of (On(A); +), consider Green's relations for the semigroup (On(A); +), characterize all constant subsemigroups of (On(A);+), all semilattices, rectangular bands and normal bands contained in (On(A);+).