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. 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. 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

   $$(\forall x)(\forall y)(\forall z)(xy=xz)\to (y=z)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}(\forall x)(\forall y)(\forall z)(yx=zx)\to (y=z).$$

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.