A Research On The Commutative Properties Of Languages

Let A be an alphabet. The language X ,Y (?) A* are called commutative if XY = YX, called partially commutative if XY (?) YX or YX ? XY. It is a well-known result that if x , y∈A+ ,xy = yx then there exist t∈A+ , m, n> 0 such that x = t m ,y = tn. If we replace the x ,y by commutative languages, the conditions are turning to be complicated.In this article, we first researched the properties of commutation or partially commutation with codes. In the aspect of the relation between commutative properties of codes and unambiguous sets, the result that if a code commutates with a certain language then their product is unambiguous is given. In the aspect of the relation between commutative properties of codes and maximal codes, the result that any code X ? A+ partially commutated by A* is a maximal code is proven. Obviously, any code X commutated by A* is a maximal code. Let X ? A+ be a code, the result are also given that X commutated by A* if and only if X = An,n is a positive integer. In the later 1980s, B. Ratoandromanana had proposed a conjecture that any code has a unique primitive root. An equivalent character of the conjecture is given; it gives a new way to resolve the conjecture.In this article, we also researched the properties of commutation with general languages. We emphatically analyzed the 1-free largest language commutated with a certain language, i.e. the centralizer of the language. We proven that there is a boundary of those finitely generated centralizers, i.e. there exists a positive integer n such that any language with elements less than or equal n has a finitely generated centralizer, and that for any positive integer m larger than n there must be a m -elements language hasn't a finitely generated centralizer. For researching the boundary problem, the utility of binary singular languages is developed. Some special cases about the finitely generated centralizer of 4 elements languages are given by using the theory mentioned above.