mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a quasivariety is a class of

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

s generalizing the notion of variety by allowing equational conditions on the axioms defining the class. __TOC__

Definition

A ''trivial algebra'' contains just one element. A quasivariety is a class ''K'' of algebras with a specified

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

satisfying any of the following equivalent conditions: # ''K'' is a pseudoelementary class closed under

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

s and

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

s. # ''K'' is the class of all

model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

s of a set of quasi-identities, that is, implications of the form

s_1 \approx t_1 \land \ldots \land s_n \approx t_n \rightarrow s \approx t

, where

s, s_1, \ldots, s_n,t, t_1, \ldots, t_n

are terms built up from variables using the operation symbols of the specified signature. # ''K'' contains a trivial algebra and is closed under

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s, subalgebras, and

reduced product In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a nonempty family of structures of the same signature σ indexed by a set ''I' ...

s. # ''K'' contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and

ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...

Examples

Every variety is a quasivariety by virtue of an equation being a quasi-identity for which . The

cancellative semigroup In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form ''a''·''b'' = ''a''·''c'', ...

s form a quasivariety. Let ''K'' be a quasivariety. Then the class of orderable algebras from ''K'' forms a quasivariety, since the preservation-of-order axioms are

Horn clause In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form that gives it useful properties for use in logic programming, formal specification, universal algebra and model theory. Horn clauses are ...

References

{{Authority control Universal algebra