In mathematics, a quasivariety is a class of

algebraic structure In mathematics, an algebraic structure 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 multiplication), and a finite set ...

s generalizing the notion of

variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...

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 la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) 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 ...

satisfying any of the following equivalent conditions. 1. ''K'' is a pseudoelementary class closed under subalgebras and direct products. 2. ''K'' is the class of all models of a set of quasiidentities, 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. 3. ''K'' contains a trivial algebra and is closed under isomorphisms, 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 family of structures of the same signature σ indexed by a set ''I'', and let ...

s. 4. ''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 factor ...

Examples

Every

is a quasivariety by virtue of an equation being a quasiidentity for which ''n'' = 0. 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'', w ...

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 which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logi ...

References

{{Authority control Universal algebra