implicational class
In this entry, we extend the notion of an equational class (or a variety^{}) 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 $\mathrm{\Sigma}$ of “equations” and that $K$ is the smallest class satisfying $\mathrm{\Sigma}$. 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 “implications^{}”, where an implication has the form $P\to Q$, where $P$ and $Q$ are some sentences^{}. Formally, we define an equational implication in an algebraic system to be a sentence of the form
$$(\forall {x}_{1})\mathrm{\cdots}(\forall {x}_{n})({e}_{1}\wedge \mathrm{\cdots}\wedge {e}_{p}\to {e}_{q}),$$ 
where each ${e}_{i}$ is an identity^{} of the form ${f}_{i}({x}_{1},\mathrm{\dots},{x}_{n})={g}_{i}({x}_{1},\mathrm{\dots},{x}_{n})$ for some $n$ary polynomials^{} ${f}_{i}$ and ${g}_{i}$, and $i=1,\mathrm{\dots},p,q$.
Definition. A class $K$ of algebraic systems of the same type (signature^{}) is called an implicational class if there is a set $\mathrm{\Sigma}$ of equational implications such that
$$K=\{A\text{is a structure}\mid A\text{is a model in}\mathrm{\Sigma}\}=\{A\mid (\forall q\in \mathrm{\Sigma})\to (A\vDash q)\}.$$ 
Examples
 1.

2.
The class of all Dedekindfinite rings. In addition to satisfying the identities for being a (unital) ring, each ring also satisfies the equational implication
$$(\forall x)(\forall y)(xy=1)\to (yx=1).$$ 
3.
The class of all cancellation semigroups. In addition to satisfying the identities for being a semigroup^{}, each semigroup also satisfies the implications
$$(\forall x)(\forall y)(\forall z)(xy=xz)\to (y=z)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}(\forall x)(\forall y)(\forall z)(yx=zx)\to (y=z).$$ 
4.
The class $K$ of all torsion free abelian groups^{}. In addition to satisfying the identities for being abelian groups, each group also satisfies the set of all implications
$$\{\forall x(nx=0)\to (x=0)\mid n\text{is a positive integer}\}.$$
There is an equivalent^{} formulation of an implicational class. Again, let $K$ be a class of algebraic systems of the same type (signature) $\tau $. Define the following four “operations^{}” on the classes of algebraic systems of type $\tau $:

1.
$I(K)$ is the class of all isomorphic^{} copies of algebras in $K$,

2.
$S(K)$ is the class of all subalgebras^{} of algebras in $K$,

3.
$P(K)$ is the class of all product^{} of algebras in $K$ (including the empty products, which means $P(K)$ includes the trivial algebra), and

4.
$U(K)$ is the class of all ultraproducts^{} of algebras in $K$.
Suppose $X$ is any one of the operations above, we say that $K$ is closed under operation $X$ if $X(K)\subseteq K$.
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 converse^{} is not true, as can be readily seen in the last example above, since a homomorphic image^{} of a torsion free abelian is in general not torsion free: the homomorphic image of $\varphi :\mathbb{Z}\to {\mathbb{Z}}_{n}$ is a subgroup^{} of ${\mathbb{Z}}_{n}$, hence not torsion free.
Title  implicational class 

Canonical name  ImplicationalClass 
Date of creation  20130322 17:31:35 
Last modified on  20130322 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 