Icombinators

Icombinators are a concept in category theory, which is a branch of mathematics. They were introduced by Gunther Schmidt in 1986 as a generalization of combinators. Combinators are functions that take other functions as arguments and return functions as output. Icombinators generalize this idea by allowing the input functions to be input-output systems, rather than just functions.

In other words, an icombinator is a function that takes systems of input-output functions as arguments and returns a new system of input-output functions as output. Icombinators are a way to compose and manipulate such systems in a more general and abstract way.

The icombinator concept is closely related to the idea of algebraic semantics and the study of the structure of categorical systems. Icombinators can be used to describe the behavior of systems and to reason about their properties. They are a useful tool in the study of category theory and its applications.

Some of the key properties of icombinators include their ability to compose and manipulate complex systems and their ability to handle non-deterministic and probabilistic computations. Icombinators have been used in a variety of fields, including concurrency theory, theoretical computer science, and logic.

Icombinators have a close relationship to functional programming languages, particularly Haskell, where they are used as a tool for describing and manipulating functions. The concept of icombinators has also led to the development of new programming paradigms and languages.

Overall, icombinators provide a powerful and expressive way to reason about systems and their behavior, and have found applications in a range of fields.