Quasiidentities

Quasiidentities are a form of logical statement used in universal algebra and model theory. They express conditional equations: a quasi-identity has the shape: for all variables, if a conjunction of equations holds, then a single equations must hold. In symbols: ∀x1,...,xn ((t1 ≈ s1 ∧ t2 ≈ s2 ∧ ... ∧ tk ≈ sk) → (t ≈ s)). If the premises are empty, a quasi-identity reduces to an ordinary equation, an identity true in every algebra of the given signature.

Quasi-identities generalize equational laws and are the building blocks of quasivarieties. A class of algebras that satisfies a given set of quasi-identities is called a quasivariety. Quasivarieties are closed under taking subalgebras, direct products, and ultraproducts, but not necessarily under homomorphic images. They correspond to the models of a universal Horn theory, meaning they are defined by Horn clauses with equality.

Examples help illustrate the idea. In the variety of abelian groups, the quasi-identity ∀x (n·x = 0 → x = 0) expresses that the group has no nontrivial n-torsion; the class of groups satisfying this quasi-identity is the quasivariety of n-torsion-free groups. In commutative rings, a related quasi-identity may assert that if elements are idempotent (x^2 = x and y^2 = y), then their product is idempotent ((xy)^2 = xy); this holds in all commutative rings and thus defines a property describable by a quasi-identity.

Overall, quasiidentities provide a flexible language for capturing conditional algebraic properties and for studying the structure and closure properties of quasivarieties.