# 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 $f \left(x_1,\dots x_ \left\{k-1\right\}, 0, x_ \left\{k+1\right\},\dots x_n\right) \ Leq F \left(x_1,\dots x_ \left\{k-1\right\}, 1, x_ \left\{k+1\right\},\dots x_n\right)$

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