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Elementary class
In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory.
== Definition ==
A class ''K'' of structures of a signature σ is called an elementary class if there is a first-order theory ''T'' of signature σ, such that ''K'' consists of all models of ''T'', i.e., of all σ-structures that satisfy ''T''. If ''T'' can be chosen as a theory consisting of a single first-order sentence, then ''K'' is called a basic elementary class.
More generally, ''K'' is a pseudo-elementary class if there is a first-order theory ''T'' of a signature that extends σ, such that ''K'' consists of all σ-structures that are reducts to σ of models of ''T''. In other words, a class ''K'' of σ-structures is pseudo-elementary iff there is an elementary class ''K''' such that ''K'' consists of precisely the reducts to σ of the structures in ''K'''.
For obvious reasons, elementary classes are also called axiomatizable in first-order logic, and basic elementary classes are called finitely axiomatizable in first-order logic. These definitions extend to other logics in the obvious way, but since the first-order case is by far the most important, axiomatizable implicitly refers to this case when no other logic is specified.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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