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.

中文翻译：

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由方程定义的代数

我们证明了一类代数在取同态图像和直接积的情况下是封闭的，当且仅当该类由满足一组（通常同时）方程的所有代数组成。特别是对于正则半群的类，这允许对通用代数性质的解释，该性质完全根据半群的结合二元运算来表述，它作为通过所谓的 e 变体的方法的替代方法。特别地，我们证明了逆半群、正统半群和 $E$-solid 半群的类在我们的意义上是等式的。