Clone (mathematics)

Clone

Definition

That is to say has a unit, and F (A) the whole of all the constants of has and of all the functions finitaires on A. a clone on has is a subset X of F (A) such as

1. X is closed by composition;

2. X contains all projections;
3. if constant C is in X, all the functions finitaires with constant value C are in X
4. if a function finitaire with constant value C is in X, then the constant value C is in X.

Any subset X of F (A) is contained in a smaller clone on has, than one calls the clone generated by X.

The clone generated by the fundamental operations of an algebra has is called the clone of A.

Example

Let us consider the whole with 2 elements 0,1. The generated clone by constants 0,1 and the binary operations min and max are that of the increasing functions, i.e. such as $f\; (x\_1,\dots \; x\_\; \{k-1\},\; 0,\; x\_\; \{k+1\},\dots \; x\_n)\; \backslash \; Leq\; F\; (x\_1,\dots \; x\_\; \{k-1\},\; 1,\; x\_\; \{k+1\},\dots \; x\_n)$

Emil Post published in 1921 an exhaustive and enthralling study of the clones on the algebras with two elements.

Random links:

Louis-Guillaume Ier de Bade | Petka (Lazarevac) | Henri Lefèvre | Tempus | Liberi Fatali | Mohammad_Fazel_Lankarani

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org