Research ArticleNo Access

Partial clones

Klaus Denecke

Institute of Mathematics, University of Potsdam, 14476 Potsdam, Karl-Liebknecht-Straße, Germany

E-mail Address: klausdenecke@hotmail.com

https://doi.org/10.1142/S1793557120501612Cited by:3 (Source: Crossref)

Abstract

A set $C$ $C$ of operations defined on a nonempty set $A$ $A$ is said to be a clone if $C$ $C$ 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 $n$ $n$ -ary operations defined on set $A$ $A$ for all natural numbers $n \geq 1$ $n \geq 1$ and the operations are the so-called superposition operations $S_{m}^{n}$ $S_{m}^{n}$ for natural numbers $m, n \geq 1$ $m, n \geq 1$ and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set $A$ $A$ 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.

Communicated by J. Koppitz

Keywords:

AMSC: 03B15, 08A30

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