The Łoś–Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formulas preserved under taking substructures is exactly the set of universal formulas.
[1] The theorem was discovered by Jerzy Łoś and Alfred Tarski.
be a theory in a first-order logic language
In more common terms, this states that every first-order formula is preserved under induced substructures if and only if it is
, i.e. logically equivalent to a first-order universal formula.
As substructures and embeddings are dual notions, this theorem is sometimes stated in its dual form: every first-order formula is preserved under embeddings on all structures if and only if it is
, i.e. logically equivalent to a first-order existential formula.
[2] Note that this property fails for finite models.
This mathematical logic-related article is a stub.