Universal Algebraic Geometry

Universal algebraic geometry over concrete algebraic structures is studied. An algebraic structure is considered and set of all simultaneous solutions of a system of equations is called the algebraic set. It is found that the category of algebraic sets over a L-structure and the category of coordinate algebras of algebraic sets are dually equivalent. Any non-empty algebraic set Y over an equationally Noetherian algebraic structure is a finite union of irreducible algebraic sets, then this decomposition is unique up to the order of the components. A structure is said to be separated by a structure if for every predicate symbol and every elements, there exists an L-homomorphism. A precise definition of direct systems and their direct limits is given using the language of diagram-formulas.