Books and Papers that have treatment of properties like Idempotence and related operations

Question

Please recommend resources to study Idempotence and other similar properties of processes and operations in depth.

I want to know what other properties like Idempotence are there for an operation. I am a computer science student so these kinds of properties of a process intrigue me.

I don't even know which field of mathematics studies these properties. Is it relations theory? abstract algebra? For example, would a good book in abstract algebra have a nice treatment of these kinds of properties?

Edit : I just want a beginner's treatment on these kinds of operation.

Answer 1

The text Introduction to Lattices and Order has material on closure operators which are idempotent and there are also some sections that deal with theoretical computer science. Here is a link to Amazon's page: http://www.amazon.com/gp/reader/0521784514/ref=sib_dp_pt#reader-link

Answer 2

I work on elementary particle theory. Calculations using "Feynman diagrams" are done using "propagators" (which are Green's functions for wave functions, i.e. solutions to the Klein-Gordon or Schroedinger or Dirac or Pauli-Dirac equation) and "vertices" (which give the amplitude for particles interacting so as to replace one group of particles with another group). An example of this sort of thing is an up quark emitting a W+ particle and becoming a down quark. One models the movement of the W+ particle with the propagator appropriate to a spin-1 massive object. The up and down quarks get modeled with spin-1/2 propagators, while the interaction gets modeled by the weak force. In short, I'd like to see this redone completely in terms of the propagators. The idea is that propagators should carry the information sufficient to describe the interaction vertices.

So to me, the propagators are very important and should be studied at the most basic level. What can be a propagator? Since a propagator describes a particles that is moving through space-time without change to the nature of the particle (that is, we use the label "particle" to describe activity that tends to preserve itself in time), there is a requirement that the propagator be idempotent. As Green's functions, you have: $$G(x,x'') = \int_{-\infty}^{+\infty}G(x,x')G(x',x'')\;dx'.$$ Eliminating the dependency on space and time gives you something that describes the particle's nature (i.e. electric charge, spin, weak hypercharge, weak isospin ..). The above becomes a matrix equation: $$G = G^2$$ Solving the above should give attributes of the elementary particles, for example see http://brannenworks.com/Gravity/qioumm_view.pdf

For this reason, you might find my (incomplete) book on Clifford algebra and elementary particles of interest as it concentrates on understanding the idempotents of the Clifford algebra: http://www.brannenworks.com/dmaa.pdf Uh, Clifford algebra is the math term for the generalization of what the physicists call the "Dirac gamma matrices". These are used to define the wave functions for spin-1/2 particles such as the electron and positron.

If you google Clifford algebra you'll find a lot of applications in computer science, so this might be of interest.

Answer 3

Trying to guess what you mean.

A definition of idempotence would be, for $f$ a function on a typical element $x$,

$$f( f(x) ) = f(x)$$

This is an example of a functional equation. So you might be interested in a book or introductory article about functional equations.

There is no limit to the number of properties you can imagine for a function based on a functional equation. Classical algebraic notions such as commutativity, distributivity, associativity can be expressed this way (see below).

Idempotence is just one of the most basic and most meaningful, related to the geometric notion of projection. Even this is too broad to find the kind of elementary treatment you want.

In general a functional equation would be any kind of equality that you can write between a function, its successive iterations and its values. You can complicate matters by introducing derivatives, auxiliary functions, several variables, going into inequations, etc.

As others have already remarked, this is a broad subject touching on very different parts of mathematics. But I would guess you are more interested on discrete objects such as integers or trees than on continuous objects.

You can at first have a look at traditional web sources such as Wikipedia, when you will find a beginning of classification. You can also look at most books or articles about solving problems at Mathematics Olympiad where it is common to have functional equations (on the sets of integers but not only).

Functional equations are used in Computer Science in different contexts such as dynamic programming, automata theory, analysis of algorithms, among others. You may have also heard about Functional Programming. In some of them basic properties of functions are used in evaluating programs and optimizations. For instance, if you can declare that a function is associative, the interpreter or compiler can choose the order in which it will evaluates the arguments since it will not change the result.