Narrow Results

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,…,g_n : Aⁿ → A are n-ary operations defined on A, then by Sⁿ(f, g₁ … , g_n)(a₁ … , a_n) := f(g₁(a₁,…,a_n),…,g_n(a₁,…,a_n)) for all a₁,…, a_n ∈ 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);+).

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, … , g_n : Aⁿ → A are n-ary operations defined on A, then by

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set W_τ(X) of all terms of type τ. If σ : {f_i | i ∈ I} → W_τ(X) is a hypersubstitution and its canonical extension, then the set

A set $C$ of operations defined on a nonempty set $A$ is said to be a clone if $C$ is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the $n$-ary operations defined on set $A$ for all natural numbers $n\ge 1$ and the operations are the so-called superposition operations $S_m^n$ for natural numbers $m,n\ge 1$ and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set $A$ and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

A term which is a formal expression defined by variables and operation symbols can be described by tree diagram. The class of terms under which the longest distance from the root to each vertex is equal is called completely expanded. In this paper, we consider the partial many-sorted operation defined on the family of all completely expanded terms of type $\tau$ and construct the partial algebra satisfying the clone axioms. The partial operations on arbitrary algebra generated by completely expanded terms and their related structures are obtained.