How can you define the notion of "state" using only a type theory language? The notion of state I'm referring to is the one described in this wikipedia page : https://en.m.wikipedia.org/wiki/State_(computer_science)

Which defines it in this way: 
 "a program is described as stateful if it is         designed to remember preceding events or user interactions; [1] the remembered information is called the state of the system."

My question is: Does there exist one or more formal definitions/descriptions that correspond to this notion of "state",  in a formal system of Type theory (might be system F , simply typed lambda calculus or any other)
I am not sure what your question exactly covers. But to have a more formal definition of "State" it could be useful to look at the State Monad in Haskell. Haskell is a functional programming language which operates different compared to a imperative programming language (like Java or C).