What is a clone (algebra)?
In algebra, a clone is a set of operations on a given set that is closed under composition and includes all projection maps.

More specifically, let S be a non-empty set and let Ar(S) denote the set of all finite sequences of elements of S. A clone on S is a set of operations on S that satisfies the following properties:

1. If f and g are operations in the clone, then the composition fg is also in the clone.
2. For every element a in S, the projection map P(a) is in the clone, where P(a)(x) = a for all x in S.

The clone generated by a subset T of S is the smallest clone containing all operations in T. The notion of a clone is important in the study of algebraic structures, particularly in universal algebra, where it is used to define and study the properties of algebraic structures such as groups, rings, and fields.
Can you give an example of a clone on a set?
Sure, here's an example of a clone on a set:

Let S be the set of real numbers, and let C be the clone generated by the set of operations {+, -, ×}, where + denotes addition, - denotes subtraction, and × denotes multiplication.

Then C is a clone on S, because it satisfies the properties of a clone:

- If f and g are operations in C, then the composition fg is also in C. For example, (x + y) - z is a composition of the addition and subtraction operations, both of which are in C, so (x + y) - z is also in C.
- For every element a in S, the projection map P(a) is in C, where P(a)(x) = a for all x in S. For example, P(1)(x) = 1 for all x in S, so P(1) is in C.

So C is a clone on S that contains the three basic arithmetic operations of addition, subtraction, and multiplication.
Who first discovered the concept of clones in algebra?  What is its history?
The concept of clones in algebra was first introduced by the American mathematician Garrett Birkhoff in 1935. Birkhoff's work on universal algebra sought to unify different branches of algebra by studying the general properties of algebraic structures.

In particular, Birkhoff introduced the concept of an algebraic clone as a class of functions that is closed under composition and contains all the projection functions. This concept provided a powerful tool for studying the algebraic properties of various structures, including groups, rings, and lattices.

Since then, the concept of clones has been further developed and applied in various areas of mathematics, including logic, complexity theory, and computer science. Clones have also been studied in relation to other mathematical concepts, such as polymorphism and Galois connections.

Today, the study of clones and related topics continues to be an active area of research in algebra and related fields.