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# Clones and Completeness

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Clones and Completeness book

# Clones and Completeness

DOI link for Clones and Completeness

Clones and Completeness book

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## ABSTRACT

In Example 1.2.14, and again in Chapters 2 and 5, we studied the concept of a clone as a set of operations defined on a base set A which is closed under composition and contains all the projection operations. In Section 10.1 we describe several ways in which we can regard a clone as an algebraic structure. In Section 10.2 we describe clones using relations, since clones occur as sets of operations which preserve relations on a set A. This description is based on a Galois-connection, with the clones corresponding to closed sets, and thus the set of all clones of operations defined on a fixed finite set forms a lattice. In Section 10.3 we shall use our results from Chapter 9 on the congruence lattice properties of two-element algebras, to give a complete description of the lattice of all clones of operations defined on a two-element set. An important question is to decide when a set of operations defined on a fixed set generates the set of all operations defined on this set, the socalled functional completeness problem. We will also describe the algebraic properties of certain classes of finite algebras connected with the functional completeness problem.