Elementary equivalence

The elementary equivalence is a term in the model theory, a branch of mathematical logic. In simple terms, the names of two structures elementarily equivalent if they satisfy the same sentences, as will be clarified below.

It was the language of first-order predicate logic with the set of symbols. Two structures and are called elementary equivalent if

For all sets, ie expressions without free variables, where the sign for "fulfilled" or " is the model of" is.

Elementary equivalent structures therefore can not be distinguished by sets of first-order predicate logic. Denoting the entirety of the theory, one can also formulate that elementary equivalent structures have the same theory.

Elementary equivalence is clearly an equivalence relation and we write if the structures are equivalent and elementary. The elementary equivalence class is - elementary, because it is characterized by the set amount of theory.

The isomorphism class of is always included in the elementary equivalence class, since isomorphic structures satisfy the same sentences. Is infinite, so this inclusion is strict, because after the Löwenheim - Skolem, there are models of different thickness, which therefore can not be isomorphic. For example, are the ordered sets and elementary equivalent to what one can easily show with the set of Fraisse, which is a purely algebraic characterization of elementary equivalence at finite set of symbols to refer to the predicate logic without reference. The disintegration of the terms isomorphism and elementary equivalence characterizes the finite models, because for a model are equivalent:

• All too elementary equivalent models are isomorphic.
• Is finite.