### Abstract

We purpose a survey on various deterministic grammars (alphabetic systems) used to define infinite words, terms, trees and graphs. The deterministic grammars over words are the morphisms which are used to generate the morphic words. The labelled deterministic grammars over words coincide with the simple (context-free) grammars or the monadic recursive schemes, and can be used to produce simple infinite trees and graphs. The deterministic grammars over terms are the recursive program schemes, and define the algebraic terms. The deterministic grammars over typed terms are used to generate a hierarchy of infinite terms and can be used to generate a hierarchy of higher order morphic words. The deterministic grammars over (finite) graphs generate the transition graphs of pushdown automata. We will present these grammars with general known results. Then we will give new results with open questions and possible extensions.