##  [Imaginary Element](/imaginary-element-0) 

 Definition

An imaginary element is an equivalence class of tuples under a definable equivalence relation, treated as a new element in the imaginary expansion M^eq of a structure; imaginaries allow quotients by definable relations to be handled as genuine sorts.

 

 

 

 

 

 





## Principle

Principle

Internalize definable quotient objects by adjoining names for equivalence classes so that definability, parameters, and automorphism actions can be discussed uniformly inside an expanded structure.

 

 

 

 

 





## Demonstration

Demonstration

If E(x,y) is a definable equivalence relation on M^n, then each class [a]_E is an imaginary; for example, cosets of a definable subgroup or the set of zeros of a polynomial modulo permutation can be represented as imaginaries in M^eq.

 

 

 

 

## Misapplication

Misapplication

Treating arbitrary set‑theoretic quotients or equivalence classes that are not definable as imaginaries, or conflating the model‑theoretic notion with unrelated uses of the word 'imaginary' (e.g., complex imaginaries).

 

 

 

 

 





## Consequence

Consequence

Adding imaginaries often simplifies classification and definability statements (e.g., makes canonical parameters actual elements), enables elimination of imaginaries analysis, and clarifies orbit/stabilizer computations under automorphism groups.

 

 

 

 

## Reversal

Reversal

The reversal is to remain in the original structure without adding sorts: instead of treating classes as elements one works with representatives and formulas about tuples, which can complicate statements about definability and invariance.

 

 

 

 

 





## Boundary

Boundary

Imaginaries are attached only to definable equivalence relations (possibly with parameters) and require the passage to M^eq or analogous expansions; arbitrary equivalence relations not definable in the language are excluded.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between keeping the universe 'concrete' (original sorts only) and enriching it with imaginaries: imaginaries simplify logic at the cost of enlarging the signature and changing what counts as an 'element'.

 

 

 

 

 





## Synthesis

Synthesis

An imaginary element is a formal device that turns a definable equivalence class into an actual element of an expanded structure, enabling uniform treatment of quotients, parameters, and invariance in model‑theoretic analysis.