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 is a class of algebraic systems satisfying a set of “equations” and that is the smallest class satisfying . 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 , where and are some sentences. Formally, we define an equational implication in an algebraic system to be a sentence of the form
Definition. A class of algebraic systems of the same type (signature) is called an implicational class if there is a set of equational implications such that
There is an equivalent formulation of an implicational class. Again, let be a class of algebraic systems of the same type (signature) . Define the following four “operations” on the classes of algebraic systems of type :
Suppose is any one of the operations above, we say that is closed under operation if .
Definition. is said to be an algebraic class if is closed under , and is said to be a quasivariety if it is algebraic and is closed under .
It can be shown that a class 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 is a subgroup of , hence not torsion free.
|Date of creation||2013-03-22 17:31:35|
|Last modified on||2013-03-22 17:31:35|
|Last modified by||CWoo (3771)|