Abstract—In this paper, decidability of the structures of real, rational, integer and natural numbers will be studied in different languages. Decidability or undecidability of mathematical structures is one of the fundamental and sometimes very difficult problems of mathematical logic, where several examples of problems in this field are still open and unresolved even after decades. One of the goals of Mathematical Logic is the axiomatization of mathematical theoriesTarski has proved the decidability of the theory of real and complex numbers in the language of addition and multiplication, and it is proved that theories of natural, integer, and rational numbers, in the language of addition and multiplication, are undecidable (Theorems of Gödel and Robinson). We will review The following problemes: The Main Problem 1: ⟨Q; ⊑⟩ is decidable? Problem 2: an explicit axiomatization for ⟨Z; \times⟩? and we will study boolean algebras. Boolean algebras are famous mathematical structures.Tarski showed the decidability of the elementary theory of Boolean algebras.In this paper, we study the different kinds of Boolean algebras and their properties. And we present for the first-order theory of atomic Boolean algebras a quantifier elimination algorithm. The subset relation is a partial order and indeed a lattice order,And I will prove that the theory of atomic Boolean lattice orders is decidable, and furthermore admits elimination of quantifiers. So the theory of the subset relation is decidable. And we will study decidability of atomlss boolean algebra.