A system of equations is a subset E of the Cartesian product X∗ × X∗ of the free monoid (finite strings) over X with itself.
A system of equations if independent if it is not equivalent to a proper subset of itself.
[1] A semigroup is compact if every independent system of equations is finite.
[2] The class of compact semigroups does not form an equational variety.
However, a variety of monoids has the property that all its members are compact if and only if all finitely generated members satisfy the maximal condition on congruences (any family of congruences, ordered by inclusion, has a maximal element).