Imaginary element

In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by , and elimination of imaginaries was introduced by .

In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by , and elimination of imaginaries was introduced by . ==Definitions== M is a model of some theory. x and y stand for n-tuples of variables, for some natural number n. An equivalence formula is a formula φ(x, y) that is a symmetric and transitive relation. Its domain is the set of elements a of M n such that φ(a, a); it is an equivalence relation on its domain. An imaginary element a/φ of M is an equivalence formula φ together with an equivalence class a. M has elimination of imaginaries if for every imaginary element a/φ there is a formula θ(x, y) such that there is a unique tuple b so that the equivalence class of a consists of the tuples x such that θ(x, b). A model has uniform elimination of imaginaries if the formula θ can be chosen independently of a. A theory has elimination of imaginaries if every model of that theory does (and similarly for uniform elimination). ==Examples== ZFC set theory has elimination of imaginaries. Peano arithmetic has uniform elimination of imaginaries. A vector space of dimension at least 2 over a finite field with at least 3 elements does not have elimination of imaginaries. ==References==

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