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An equational class is a set K of similar mathematical models (i.e. they have the same language) that satisfies certain equations. In other words we have some set of equations Eq and if some structure M satisfies all the equations in Eq then M is an element of K.

Ok, let's get down to some examples... Firstly we take a look at groups. (Recall that a group is an algebra with one binary operation `+', a unary operation `-' and a constant `0'.) Then the equational class of all groups is determined by the following equations (these are the equations that will be in the set Eq)

To get the equational class of Abelian groups we add the following equation to those above In essence we are saying that we are cutting out those groups that are non-commutative from the equational class of groups to get the Abelian groups.

For a final example we take a look at Boolean algebras. (Recall that a Boolean algebra has two binary operations `v' (and) and `^' (or), a unary operation `~' (not) and two constants `0' (false) and `1' (true)). The equations characterising the equational class of Boolean algebras is then as follows.

  • Xv(YvZ)=(XvY)vZ
  • Xv(~X)=1
  • Xv1=1
  • XvY=YvX
  • Xv(Y^Z)=(XvY)^(XvZ)    (distributivity)
  • XvY=(~X)^(~Y)    (De Morgan).
Comparing the first four equations here to those defining the equational class of Abelian groups above we see that every Boolean algebra is also a Abelian group. (Strictly speaking this is not true since the signature of Boolean algebras contains more symbols than the one for groups and hence they are completely different kinds of models.)

One of the most central theorems about these classes Birkhoff's Theorem states that each equational class is a variety.

These classes generally arise in the fields of model theory, universal algebra and mathematical logic.

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