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# 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. 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. 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. 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. $I(K)$ 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 $\phi:\mathbb{Z}\to\mathbb{Z}_{n}$ is a subgroup of $\mathbb{Z}_{n}$, hence not torsion free.

## Mathematics Subject Classification

08C15*no label found*03C05

*no label found*

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