A short review of decidability of boolean algebras ‎and‎ structure‎‏e of rational numbers ‎in different languages

This article consists of two parts.First,‎‎we study boolean algebras.Boolean algebras are famous mathematical structures.Tarski showed the decidability of the elementary theory of Booleana lgebras.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 isdecidable‎.‎‎And ‎we ‎will ‎study decidability of atomlss boolean algebra.Second part,‎of this paper we sho‏w that the structure of rational numbers in different languages ​​has the property of ‎quantifier ‎elimination,and hence is decidable.This proofes are organized in two‎ parts.