implicational class

In this entry, we extend the notion of an equational class (or a varietyMathworldPlanetmath) to a more general notion known as an implicational class (or a quasivariety). Recall that an equational class K is a class of algebraic systems satisfying a set Σ of “equations” and that K is the smallest class satisfying Σ. Typical examples are the varieties of groups, rings, or lattices.

An implicational class, loosely speaking, is the smallest class of algebraic systems satisfying a set of “implicationsMathworldPlanetmath”, where an implication has the form PQ, where P and Q are some sentencesMathworldPlanetmath. Formally, we define an equational implication in an algebraic system to be a sentence of the form


where each ei is an identityPlanetmathPlanetmathPlanetmathPlanetmath of the form fi(x1,,xn)=gi(x1,,xn) for some n-ary polynomialsMathworldPlanetmath fi and gi, and i=1,,p,q.

Definition. A class K of algebraic systems of the same type (signaturePlanetmathPlanetmath) is called an implicational class if there is a set Σ of equational implications such that

K={A is a structure A is a model in Σ}={A(qΣ)(Aq)}.


  1. 1.

    Any equational class is implicational. Each identity p=q can be thought of as an equational implication (p=p)(p=q). In other words, every algebraMathworldPlanetmathPlanetmath satisfying the identity also satisfies the corresponding equational implication, and vice versa.

  2. 2.

    The class of all Dedekind-finite rings. In addition to satisfying the identities for being a (unital) ring, each ring also satisfies the equational implication

  3. 3.

    The class of all cancellation semigroups. In addition to satisfying the identities for being a semigroupPlanetmathPlanetmath, each semigroup also satisfies the implications

  4. 4.

    The class K of all torsion free abelian groupsMathworldPlanetmath. In addition to satisfying the identities for being abelian groups, each group also satisfies the set of all implications

    {x(nx=0)(x=0)n is a positive integer}.

There is an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath formulation of an implicational class. Again, let K be a class of algebraic systems of the same type (signature) τ. Define the following four “operationsMathworldPlanetmath” on the classes of algebraic systems of type τ:

  1. 1.

    I(K) is the class of all isomorphicPlanetmathPlanetmathPlanetmath copies of algebras in K,

  2. 2.

    S(K) is the class of all subalgebrasPlanetmathPlanetmath of algebras in K,

  3. 3.

    P(K) is the class of all productMathworldPlanetmath of algebras in K (including the empty products, which means P(K) includes the trivial algebra), and

  4. 4.

    U(K) is the class of all ultraproductsMathworldPlanetmath of algebras in K.

Suppose X is any one of the operations above, we say that K is closed under operation X if X(K)K.

Definition. K is said to be an algebraic class if K is closed under I, and K is said to be a quasivariety if it is algebraic and is closed under S,P,U.

It can be shown that a class K of algebraic systems of the same type is implicational iff it is a quasivariety. Therefore, we may use the two terms interchangeably.

As we have seen earlier, a variety is a quasivariety. However, the converseMathworldPlanetmath is not true, as can be readily seen in the last example above, since a homomorphic imagePlanetmathPlanetmathPlanetmath of a torsion free abelian is in general not torsion free: the homomorphic image of ϕ:n is a subgroupMathworldPlanetmathPlanetmath of n, hence not torsion free.

Title implicational class
Canonical name ImplicationalClass
Date of creation 2013-03-22 17:31:35
Last modified on 2013-03-22 17:31:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 08C15
Classification msc 03C05
Synonym quasivariety
Synonym quasiprimitive class
Defines algebraic class
Defines equational implication