Quasivarieties

A quasivariety is a concept in universal algebra and logic that generalizes the notions of varieties and quasivarieties. It is a class of algebraic structures of a fixed signature that is closed under subalgebras, direct products, and ultraproducts, but not necessarily under arbitrary quotient structures. Quasivarieties are often characterized by sets of quasi-identities, which are implications between conjunctions of equations.

Formally, a class of algebras is called a quasivariety if it can be defined by a set of quasi-identical sentences (rules of the form "if these equations hold, then this equation holds") within a given algebraic language. Quasivarieties are important because they provide a framework for studying classes of algebraic structures that are not necessarily varieties, thus allowing the inclusion of more general logical and algebraic properties.

An essential property of quasivarieties is that they are closed under subalgebras, direct products, and ultraproducts, but not necessarily under homomorphic images. This distinguishes them from varieties, which are also closed under homomorphic images. Quasivarieties often arise in the study of logical systems, model theory, and algebraic structures where certain formal implications are specified.

The theory of quasivarieties provides tools for understanding the structure and classification of algebraic systems, especially those that obey certain conditional identities. They serve as a unifying framework bridging algebra, logic, and model theory, offering insights into the relationships between logical definability and algebraic properties.