  We show that a class of algebras is closed under the taking of homomorphic
images and direct products if and only if the class consists of all algebras
that satisfy a set of (generally simultaneous) equations. For classes of
regular semigroups in particular this allows an interpretation of a universal
algebraic nature that is formulated entirely in terms of the associative binary
operation of the semigroup, which serves as an alternative to the approach via
so called e-varieties. In particular we prove that classes of Inverse
semigroups, Orthodox semigroups, and $E$-solid semigroups are equational in our
sense.
