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.
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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
, where
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 ...
s.
Examples
Every
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), ...
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 ...
s.
References
{{Authority control
Universal algebra