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 $\Sigma$ of “equations” and that $K$ is the smallest class satisfying $\Sigma$. 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

 $(\forall x_{1})\cdots(\forall x_{n})(e_{1}\wedge\cdots\wedge e_{p}\to e_{q}),$

where each $e_{i}$ is an identity of the form $f_{i}(x_{1},\ldots,x_{n})=g_{i}(x_{1},\ldots,x_{n})$ for some $n$-ary polynomials $f_{i}$ and $g_{i}$, and $i=1,\ldots,p,q$.

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

 $K=\{A\mbox{ is a structure }\mid A\mbox{ is a model in }\Sigma\}=\{A\mid(% \forall q\in\Sigma)\to(A\models q)\}.$

Examples

1. 1.

Any equational class is implicational. Each identity $p=q$ can be thought of as an equational implication $(p=p)\to(p=q)$. In other words, every algebra 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

 $(\forall x)(\forall y)(xy=1)\to(yx=1).$
3. 3.

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

 $(\forall x)(\forall y)(\forall z)(xy=xz)\to(y=z)\quad\mbox{and}\quad(\forall x% )(\forall y)(\forall z)(yx=zx)\to(y=z).$
4. 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

 $\{\forall x(nx=0)\to(x=0)\mid n\mbox{ is a positive integer}\}.$

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. 1.

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

2. 2.

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

3. 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. 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 $\phi:\mathbb{Z}\to\mathbb{Z}_{n}$ is a subgroup of $\mathbb{Z}_{n}$, hence not torsion free.

Title implicational class ImplicationalClass 2013-03-22 17:31:35 2013-03-22 17:31:35 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 08C15 msc 03C05 quasivariety quasiprimitive class algebraic class equational implication