Generalization of the Theory of Finite Semigroup Automata

Observing the automata, we can see that they, during their work, make a transition from one state to another, on which a word from an alphabet corresponds, and in the end they finish in some final states. Here, the question how to define automata is posed, which automata will have words from an alphabet for an input, while they finish in some final states and words from the alphabet on which they are treated. Namely, every transition from one state to another will correspond with the words from the given alphabet which will be mapped to words with smaller length.
In the talk I will give a generalization of finite semigroup automata and
languages to the vector valued semigroup automata and vector valued languages.
Also, the Eilenberg's theorem for the (3,2)-automata will be discussed.