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In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences with parameters from the structure that are true in the structure. It is also called the complete diagram .
This page focuses on finitary first order model theory of infinite structures.. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:
In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others.
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x 1 , x 2 ,..., x n that are true of a set of n -tuples of an L ...
The theory of finite groups is the set of first-order statements in the language of groups that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or ...
Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if T is an -categorical theory, then it always has a model companion. [1] [2] A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the diagram ...
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe.
In model theory, the case of being algebraically closed and its prime field is especially important. While vector spaces are modular and affine spaces are "almost" modular (i.e. everywhere locally modular), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular).