mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 of ...

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 handwritten (and often 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 intent. The writer of a ...

satisfying any of the following equivalent conditions. 1. ''K'' is a

pseudoelementary class In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory ...

closed under subalgebras and direct products. 2. ''K'' is the class of all models of a set of

quasiidentities In universal algebra, a quasi-identity is an implication of the form :''s''1 = ''t''1 ∧ … ∧ ''s'n'' = ''t'n'' → ''s'' = ''t'' where ''s''1, ..., ''s'n'', ''t''1, ..., ''t'n'', ''s'', and ''t'' are terms built up from variable ...

, 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

term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...

s 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 factors ...

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'', wh ...

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 log ...

References

{{Authority control Universal algebra